The Role of Extracellular Conductivity Proﬁles in ...

LETTER Communicated by Alain Destexhe

The Role of Extracellular Conductivity Profilesin Compartmental Models for Neurons:Particulars for Layer 5 Pyramidal Cells

Kai WangDepartment of Functional Brain Imaging, Institute of Development,Aging, and Cancer, Tohoku University, Aoba-ku Sendai 980-8575, Japan

Jorge [email protected] of Biomedical Engineering, Florida International University,Miami, FL 33174, U.S.A., and Department of Functional Brain Imaging,Institute of Development, Aging, and Cancer, Tohoku University,Aoba-ku Sendai 980-8575, Japan

Herve Enjieu-KadjiDepartment of Functional Brain Imaging, Institute of Development, Aging,and Cancer, Tohoku University, Aoba-ku Sendai 980-8575, Japan, andMonell Chemical Senses Center, Philadelphia, PA 19104, U.S.A.

Ryuta KawashimaDepartment of Functional Brain Imaging, Institute of Development, Aging,and Cancer, Smart Ageing International Research Center, TohokuUniversity, Aoba-ku Sendai 980-8575, Japan

With the rapid increase in the number of technologies aimed at observ-ing electric activity inside the brain, scientists have felt the urge to createproper links between intracellular- and extracellular-based experimentalapproaches. Biophysical models at both physical scales have been formal-ized under assumptions that impede the creation of such links. In thiswork, we address this issue by proposing a multicompartment model thatallows the introduction of complex extracellular and intracellular resis-tivity profiles. This model accounts for the geometrical and electrotonicproperties of any type of neuron through the combination of four devices:the integrator, the propagator, the 3D connector, and the collector. In par-ticular, we applied this framework to model the tufted pyramidal cellsof layer 5 (PCL5) in the neocortex. Our model was able to reproduce thedecay and delay curves of backpropagating action potentials (APs) in this

∗Correspondence to Jorge Riera.

Neural Computation 25, 1807–1852 (2013) c© 2013 Massachusetts Institute of Technology

1808 K. Wang, J. Riera, H. Enjieu-Kadji, and R. Kawashima

type of cell with better agreement with experimental data. We used thevoltage drops of the extracellular resistances at each compartment to ap-proximate the local field potentials generated by a PCL5 located in closeproximity to linear microelectrode arrays. Based on the voltage dropsproduced by backpropagating APs, we were able to estimate the currentmultipolar moments generated by a PCL5. By adding external currentsources in parallel to the extracellular resistances, we were able to createa sensitivity profile of PCL5 to electric current injections from nearbymicroelectrodes. In our model for PCL5, the kinetics and spatial profileof each ionic current were determined based on a literature survey, andthe geometrical properties of these cells were evaluated experimentally.We concluded that the inclusion of the extracellular space in the compart-mental models of neurons as an extra electrotonic medium is crucial forthe accurate simulation of both the propagation of the electric potentialsalong the neuronal dendrites and the neuronal reactivity to an electricalstimulation using external microelectrodes.

1 Introduction

Neuronal activity at the cellular level has been investigated in the past us-ing two electrophysiological approaches. In the first one, individual neuronsare targeted in situ or in vitro with glass microelectrodes using a variety ofrecording and preparation protocols. Whole-cell voltage and current clamp-ing is considered the most useful recording protocol and is typically per-formed on acute slice preparations (Neher, 1971; Sakmann & Neher, 1984;Stuart & Sakmann, 1995; Angelo, London, Christensen, & Hausser, 2007;Bar-Yehuda, Ben-Porat, & Korngreen, 2008). Extracellular recordings per-formed in vivo using microelectrode arrays (MEAs) constitute the secondand the earliest approach, dating from the work of Emil Heinrich du Bois-Reymonds, who introduced nonpolarizable electrodes and high-sensitivitymultipliers in the mid nineteenth century (Pearce, 2001). Currently, extra-cellular recordings in experimental animals and human patients via in-tracranial windows are a daily practice in many research institutes andhospitals worldwide (Brinkmann, Bower, Stengel, Worrell, & Stead, 2009;Gnatkovsky, Librizzi, Trombin, & de Curtis, 2008; Wilent et al., 2011). Thistechnique has been divided into two fields of study: one dedicated to theunderstanding of postsynaptic potentials (i.e., local field potentials, LFPs)(Buzsaki, 2006) and the other dedicated to the genesis of neuronal spik-ing (i.e., multi(single) unit activity, M(S)UA) (Stark & Abeles, 2007; Wilson,2010).

The development of biophysical models, with particulars for each tech-nique, to explain the data has been an important issue in the history of thesetwo experimental approaches. The existence of incompatible specificities,

Modeling Conductivity Profiles of PCs 1809

together with the spontaneous segregation of electrophysiologists into twoindependent research communities, has gradually created a gap betweenthe theoretical frameworks underlying these approaches. The clearest oneis the assumption of an extracellular space with a resistance that is negli-gible with respect to the resistance of the intracellular space when creatingmodels for individual neurons from whole-cell voltage and current clamp-ing data (Rall, 1957, 1959, 1960, 1964). This assumption originated perhapsfrom the fact that the intracellular space along dendritic branches alwayscontains long and narrow domains, which afford them a very large effec-tive resistance compared with that of the extracellular space. In contrast,the volume fraction of the extracellular and intracellular spaces is about 0.3in most brain tissue preparations (Lehmenkuhler, Sykova, Svoboda, Zilles,& Nicholson, 1993), a fact that implies that the extracellular resistance is,in the extreme case, 1.3 times higher than the intracellular resistance (seeappendix 1). From an experimental perspective, whole cell voltage andcurrent clamping is the result of the observation of either induced volt-age differences or current flows between a glass microelectrode, the tipof which is placed inside the cell, and a bath electrode located far away.In contrast, extracellular recordings capture voltage differences in the ex-tracellular space from microelectrodes that are placed in close proximityto the neurons. The development of silicon-based technologies and micro-electromechanical systems (MEMS) allows the everyday building of MEAswith more precision, which has triggered remarkable advances in modelingand methods of data analysis. Current source density (CSD) analyses basedon both models of cortical columns and the Poisson equation for the elec-tric potentials constitutes one of the standard techniques used currently toanalyze LFPs (Somogyvari, Zalanyi, Ulbert, & Erdi, 2005; Pettersen, Devor,Ulbert, Dale, & Einevoll, 2006; Linden, Pettersen, & Einevoll, 2010; Einevoll,2010). Similarly, the methods used to detect and classify neuronal spikingare progressively founded on biophysical models of single neurons actingin a highly conductive medium (Gold, Henze, Koch, & Buzsaki, 2006; Rieraet al., 2012).

Recently, several groups have established techniques to perform whole-cell current clamp recordings (or juxtacellular recordings) in vivo (Joshi &Hawken, 2006; Pinault, 2008). In more ambitious projects, these recordingshave been observed simultaneously with LFPs and M(S)UA from MEA, ei-ther in situ (Gloveli et al., 2005) or in vivo (Harris, Henze, Csicsvari, Hirase,& Buzsaki, 2000; Henze et al., 2000) situations. Although each theoreticalframework referred to above has always been consistent with the respec-tive experimental approach, they should be used carefully to explain dataconcurrently observed with these different approaches. From a modelingviewpoint, the classic way to link intracellular and extracellular recordingmodalities is by solving the respective forward generative problems usinga sequential strategy. In the first step of this strategy, multicompartmental


models, which are useful to describe membrane potentials, are created withparticularities for each type of neuron. In a second step, primary and return-ing current sources across the entire membranes of neurons are calculatedusing these models. The validity of this type of source model rests onthe assumption of a space-shunted extracellular space. Finally, the Poissonequation is used to calculate the distributions of electric potentials in theextracellular space created by these transmembrane current sources (Goldet al., 2007). The major contradiction in this strategy is the fact that thefirst step is performed under the assumption of a zero extracellular resis-tance, whereas the second step is, by principle, based on the existence ofan extracellular electric conductivity profile that is quite different from zero(Pettersen, Devor, Ulbert, Dale, & Einevoll, 2006; Goto et al., 2010). Thisstrategy might be valid and useful in situations where the electrodes usedto record the extracellular potentials are placed far away from the neuronalpopulations of interest. However, in our opinion, when these electrodesare immersed inside these neuronal populations and are, hence, in veryclose proximity to the neurons generating the extracellular potentials, othertheoretical frameworks may be more appropriate. Past attempts to intro-duce extracellular resistances in compartmental models of neurons werebased on a strategy that comprises (Rall, 1962; Tuckwell 1988; Johnston &Wu, 1994) the introduction of dimensionless distance and time variablesin the cable equation, the linearization of ionic current kinetics inside eachdendritic branch, and the use of the equivalent cylinder theorem for den-dritic trees (i.e., determining input resistances for branches and dendriticattenuation effects). The last is formulated on the basis of three main con-ditions: (1) the cumulative electrotonic length condition, (2) the 3/2 powerlaw at every branch point condition, and (3) the termination condition.In this theory, the extracellular resistances are included in the length con-stant for each dendritic section, λ = √

rm/(ri + re). In contrast, Riera, Wan,Jimenez, and Kawashima (2006) proposed a different strategy to obtain amulticompartmental model for the pyramidal cells (PCs) located in the hu-man visual cortex, which explicitly incorporates extracellular resistances inall compartments using Kirchhoff’s laws for a complex electronic circuit ofmultiple branches. The interactions between compartments were only elec-trotonic (i.e., no active ionic conductances) in this preliminary work, andonly three representative compartments were considered to describe thesoma, as well as the apical-tufted and basal dendrites. The authors used thevoltage differences along the extracellular resistances to model mesoscopicdipolar sources underlying the electroencephalography (EEG) recordings(Riera, Jimenez, Wan, Kawashima, & Ozaki, 2007).

In this letter, we created a new modeling framework to simulate theelectrical activity of different neurons. This framework was inspired by aprevious study performed by Riera et al. (2006). This model is based on fourconstitutive electrical devices for most of the typical cellular structures:the collector (e.g., the soma), the propagator (e.g., the trunk of PCs), the


integrator (e.g., a dendritic branch), and the 3D connector (e.g., a dendriticbifurcation). The innovative aspect of these devices is the inclusion of bothmultiple extracellular resistances that can be used to absorb the geometricalaspects of each cellular structure. First, we created the theoretical equationsand provided instructions on how to use the four devices to create neu-rons with dissimilar morphologies. Second, we developed the particularsfor the tufted PCs of layer 5 (PCL5) in the neocortex. This constitutes analternative approach to model PCL5 that is clearly different from those de-veloped previously (Stratford, Mason, Larkman, Major, & Jack, 1989; Lytton& Sejnowski, 1991; Bush & Sejnowski, 1993, Destexhe, 2001). We used thismodel to study the effect of the extracellular resistances on action potential(AP) backpropagation along PCs. The kinetics and permeability profiles ofthe most important ionic channels in this cell type were defined based onan updated revision of the literature. Our model was able to accommodateexperimental data about the amplitude decay and peak delay of backprop-agating APs more precisely than was the same model when all extracellularresistances were set to zero. The extracellular potentials near a particularneuron were defined as the voltage drops in the extracellular resistances.We used such a construct to create individual CSDs for backpropagatingAPs and discussed our results together with experimental data reported inthe literature (Bereshpolova, Amitai, Gusev, Stoelzel, & Swadlow, 2007). Wewere able to estimate close-field monopolar, dipolar, and quadrupolar con-tributions to the CSD by a single PCL5. We concluded that even for modelsbased on Kirchhoff’s laws for circuit loops, monopolar and quadrupolarsources emerge in the CSD analysis as a result of the mismatch betweenthe physical assumptions made to solve the forward problems in these twotissue substrates: we used the quasi-static approach to determine the ex-tracellular electric fields produced by a neuron (Plonsey & Heppner, 1967)and introduced highly dispersive elements (e.g., membrane capacitances)while describing its membrane electric potential. Finally, we extended ourmodel of the PC to include elemental current sources in parallel with theextracellular resistances for each compartment, which was very useful tomimic current stimulation by microelectrodes placed in close proximity tothe neurons. Using the latter model, we were able to evaluate the sensitiv-ity of PCL5 to electric current stimulation with microelectrodes placed atdifferent locations along the cellular trunk.

2 Materials and Methods

2.1 A General Framework for Modeling Neuronal Activity. In the pro-posed general framework, neurons can be approximated by complex arrayswith four elemental building blocks (see Figures 1A.2 and 1B.2), which canin general be endowed with ionic channels. These building blocks, which aretermed integrator (dendritic branches), propagator (long trunks), 3D con-nector (branch bifurcation points), and collector (somas), contain detailed


Figure 1: General multicompartmental model for neurons. (A1) Morphometryof a PCL5 neuron. (A2) Devices of the model of PCL5 neurons. (B1) Morphome-try of a spiny stellate neuron. (B2) Devices of the model of spiny stellate neurons.In panels A2 and B2, the letters I, P, 3D, and C denote the integrator, propagator,3D connector, and collector, respectively. The integrator is defined as a devicethat possesses many tuft branches with heterogeneous impedances, receives ahuge amount of input voltages, and produces a single output voltage in the lastcompartment of the integrator, which is used as the input into another buildingblock. The propagator refers to a device that propagates APs. The propagator iscomposed of several compartments that are, in principle, nonuniform. The 3Dconnector is useful to connect three other electrotonic devices. The collector rep-resents the device that collects different inputs. Figure 1B.1 has been modifiedfrom da Costa & Martin (2011).


information about neuron geometries in terms of the particular values ofintracellular, extracellular, and membrane electrotonic parameters (i.e., re-sistances and capacitances). For instance, the pyramidal neurons (see Figure1A.1) are approximated by a collecting soma that is attached to the inte-grating basal dendrites on one side and a long propagating trunk on theother (see Figure 1A.2). The trunk ends on a bulk of integrating apical den-drites. We can also attach integrating oblique dendrites to the trunk usinga 3D connector. In contrast, spiny stellate neurons (see Figure 1B.1) can becreated by connecting several integrators to a single collector in a sphericalarray (see Figure 1B.2).

As shown in Figures 1A.2 and 1B.2, the integrator is defined as a devicethat contains two parts. The first part possesses multiple dendritic brancheswith heterogeneous impedances and receives synaptic inputs. The secondpart produces a single output voltage at the last compartment of the inte-grator, which is used as an input in any other building block connected toit. The propagator refers to a device that propagates electric potentials overlong distances separating two building blocks. It is composed of severalcompartments that can be nonuniform. Using a 3D connector, we can createlinks between three particular building blocks simulating points of electro-tonic division (e.g., dendritic bifurcations). Finally, the collector representsa device that collects the outputs from specific arrays of building blocks(i.e., the soma) and generates the final spiking state of the cell from theseoutputs.

The extracellular space (ECS) is a narrow interstitial space located be-tween the processes of neurons and glial cells. Its chemical composition issimilar to that of the cerebral spinal fluid and different from that of the intra-cellular space (ICS). It is well established that the ECS/ICS volume fractionvaries between 0.15 and 0.30 in normal brain tissues (Sykova & Nicholson,2008). The equivalent electrical circuits of the building blocks in the fre-quency domain are shown in Figures 2A, 2B, 2C, and 2D for the integrator,propagator, 3D connector, and collector, respectively. Every equivalent elec-trical circuit contains the resistances of both the ICS and ECS, as well as themembrane-complex resistance. Using Kirchhoff’s laws in these circuits, wewere able to obtain theoretical formulas for the changes in the membranepotentials for each building block (see appendix 2).

2.2 An Application: AP Backpropagation in PCL5.

2.2.1 Morphological Properties. As an example, we used this generalframework to model backpropagating APs in PCL5 (see Figure 1A.1). Inthis model, the PCL5 is composed of six interconnected building blocks(see Figure 3, left) embedded with ionic channels: three integrators (basal,oblique, and apical dendrites), a propagator (trunk), a 3D connector, and acollector (soma).


A

B

C

D


In the case of stimulation of a PCL5 via the injection of intracellularcurrents into the soma, the profiles of ion channels located along the trunkand apical dendrites play a key role in backpropagating APs. We simplifiedthe apical dendrites as an integrator with a single branch equivalent to theentire apical dendritic tree of PCL5. In the same way, the oblique and basaldendrites were modeled as equivalent single-branch integrators connectedto the trunk and the soma, respectively. The electrotonic parameters forthese single-branch integrators are calculated in appendix 3 based on datareported by Romand, Wang, Toledo-Rodriguez, and Markram (2011). There-fore, this simple model of PCL5 contained an apical integrator with only onetufted branch, a propagator with 10 compartments, an oblique integratorwith one branch attached to a 3D connector, and a collector including one in-put coming from the propagator and another coming from a basal integratorof a single branch. The 3D connector was linked to the propagator betweenthe nineth and tenth compartments (very close to the soma). We assumedthat all compartments of the propagator can be divided into two equal partsin the electrical circuit (see Figure 2B): the upper and lower parts. Note thatthe distal apical dendritic integrator was connected to the first compartmentof the propagator and that the collecting soma was connected to the lastcompartment of the propagator. The PCL5 model was programmed in Mat-lab. (A documented toolbox comprising all codes used in this study is avail-able online at http://www.idac.tohoku.ac.jp/bir/en/db/rb/101028.html.)To evaluate the performance of the proposed model, we created simula-tions of backpropagating APs along the trunk of PCL5 using the standard

Figure 2: Equivalent electrical circuits of the general multicompartmentalmodel. Equivalent electrical circuits for the integrator (A), the propagator (B),the 3D connector, (C) and the collector (D). The light gray rectangle depicts theresistance of the ICS, the dark gray rectangle depicts the resistance of the ECS,and the black rectangle depicts the membrane resistance. The electromotiveforces are caused by the ionic currents across the membrane. (A) The electri-cal circuit of the integrator contains two parts. The first part possesses manytufted branches with heterogeneous impedances and receives a huge amountof inputs, whereas the second part produces a single output voltage in thelast compartment of the integrator, to be used as an input into another device.(B) Electrical circuit of the propagator. The two inputs to this device can comefrom any other electrotonic device. The electrical circuit of the propagator iscomposed of several compartments, which have all been divided into two parts:the upper part and the lower part. (C) Electrical circuit of the 3D connector, adevice that serves to connect three other electrotonic devices. (D) Electricalcircuit of the collector, which collects several inputs that result from the out-puts of other electrotonic devices. The mathematical equations that describethe dynamics of electric potentials in these devices are shown in appendix 2(integrator, A; propagator, B; 3D connector, C; collector, D).


A

CCIb

P

3D

P

Io

Ia

B2

B1

Figure 3: A simple model for AP backpropagation in PCL5. (Left) A simplemodel of PCL5 comprising three integrators with single branches: Ib, basal; Io,oblique; Ia, apical dendrites; P, a propagator made of 10 compartments; C , acollector that has inputs from both one of the integrators (e.g., basal dendrites)and the propagator. (Right) A: The propagator of the PCL5 neuron has 10regions of interest (ROIs), termed compartments. B1: Every compartment ofthe propagator was approximated to a slender body with a particular diameterand length. B2: The collector was approximated to a sphere with a particulardiameter.

software NEURON (http://www.neuron.yale.edu/neuron/) with a sin-gle layer for the extracellular space (nlayer = 2, xc = 0 μF/cm2, and xg =0 mho/cm2; this NEURON code is also available at our laboratory’s website,provided above). The axial extracellular resistance is defined in appendix 1.The ion channel kinetics of PCL5 were defined as for deep neocortical PCs(ModelDB, NEURON). In particular, we selected the model proposed byKeren, Peled, and Korngreen (2005). We used the same morphometric pa-rameters and spatial conductivity profiles as defined in our model. We as-sumed that the ECS/ICS volume fraction in close proximity to the PCL5 wasequal to 0.30, which represents an upper bound for extracellular resistance.We used geometrical parameters to calculate the membrane resistances, aswell as the resistance of the ICS and ECS for every compartment of theintegrators, the propagator, and the collector of our model. The calculationof these resistances is shown in appendix 1.

To estimate the geometrical properties (length, diameter, area, and vol-ume) of PCL5, we performed whole-cell somatic patch clamp experiments


using somatosensory coronal slices (300 μm) from young Wistar rats(postnatal day 14 (P14) to P16). The artificial cerebral spinal fluid so-lution contained (in mM) 125 NaCl, 25 NaHCO3, 25 glucose, 2.5 KCl,1.25 NaH2PO4H2O, 2CaCl22H2O, and 1 MgCl26H2O. Recording electrodes(5–7.5 M�) were loaded with intracellular solution containing (in mM) 115potassium gluconate, 20 KCl, 2 Mg ATP, 2 Na2 ATP, 10 sodium phospho-creatine, 0.3 GTP, 10 HEPES, and 0.05 Alexa Fluor 594 Hydrazide. SixteenPCL5 were selected based on their firing patterns (i.e., regular spiking), andimage stacks recorded in these cells using a multiphoton laser scanning mi-croscopy were combined using a volume integration and alignment system(VIAS) (Rodriguez et al., 2003). The geometrical properties of selected PCL5were evaluated using the Neuron Studio software (Rodriguez et al., 2003).The morphological properties of the single branch of the apical integratorwere obtained based on averaged data from three branches, which wereused to create the equivalent last compartment of the integrator. The propa-gator was assumed to be a cylinder and was divided into 10 compartments(see Figure 3, right A), the lengths and diameters of which were determined(Figure 3, right B1). The collectors were approximated by spheres, the diam-eters of which were measured (see Figure 3, right B2). As a consequence oftechnical limitations, we were unable to reconstruct the geometrical prop-erties of the oblique and basal dendrites; hence, they were defined based ona previous study performed by Romand et al. (2011). All animal procedureswere reviewed and approved by the Tohoku University Animal StudiesCommittee.

2.2.2 Voltage-Gated Ion Currents. Four voltage-gated ion channels wereintroduced in this model: sodium (Na), potassium fast (Kf), potassium slow(Ks), and hyperpolarized-activated (h) channels. The kinetics of the firstthree ionic conductances were based on nucleated patched recordings fromPCL5 (Korngreen & Sakmann, 2000). The last ionic conductance was basedon cell-attached patch-clamp recordings from PCL5 (Kole, Hallermann, &Stuart, 2006). All these conductances were modeled using the Hodgkin-Huxley formalism (Hodgkin & Huxley, 1952).

The kinetic equations for Na, Kf, and Ks channels were defined as follows(Keren et al., 2005).

1. Sodium (Na)

gNa = gNam3h

Activation: m∞ = 1

1 + e− Vm+38

10

τm = 0.058 + 0.114e−( Vm+36

28

)2

Inactivation: h∞ = 1

1 + eVm+66

6

τh = 0.28 + 16.7e−( Vm+60

25

)2


2. Potassium fast (Kf)

gK f = gK f a4b

Activation: a∞ = 1

1 + e− Vm+47

29

τa = 0.34 + 0.92e−( Vm+71

59

)2

Inactivation: b∞ = 1

1 + eVm+66

10

τb = 8 + 49e−( Vm+73

23

)2

3. Potassium slow (Ks)

gKs = gKsγ2(0.5s1 + 0.5s2)

Activation: γ∞ = αγ

αγ + βγ

τγ = 1αγ + βγ

αγ = 0.0052(Vm − 11.1)

1 − e− Vm−11.1

13.1

βγ = 0.02e− Vm+1.27

71 − 0.005

Inactivation: s1,∞ = s2,∞ = 1

1 + eVm+58

11

τs1 = 360 + [1010 + 23.7(Vm + 54)]e−( Vm+75

48

)2

τs2 = 2350 + 1380e−0.011Vm − 210e−0.03Vm

The kinetic equations for h channels were defined as follows (Kole et al.,2006):

4. Hyperpolarized activated (h)

gIh = gIho

αo = 6.43(Vm + 154)

e( Vm+154

11.19 −1) βo = 193e

Vm33.1

o∞ = αo

αo + βoτo = 1

αo + βo

The conductance profile of each of these four channels along the trunkof the PCL5 was based on data provided by Rhodes (2006) and Kole et al.(2006). As the kinetic models proposed in Rhodes (2006) for the Na and K(fast and slow) channels were based on data recorded in previous studiesusing 40-day-old Wistar rats, the same profiles were scaled by a factor of 0.5to represent the respective conductance profiles for juvenile rats (P14–16).


This factor was described previously for the potassium channels (K f and Ks)

(Schaefer et al., 2007). We assumed that an equivalent scaling factor mightalso exist for the sodium channel (Na), which may explain the differencesin the conductance values for this channel reported by Traub et al. (2005)and Rhodes (2006) in the somas of intrinsic bursting PCL5.

We did not include calcium channels for two main reasons: (1) the spatialprofiles of this type of ionic conductance have not been reported for corticalPCs, and (2) we used our model to understand backpropagation mecha-nisms, to which calcium channels do not contribute significantly (Larkum,Nevian, Sandler, Polsky, & Schiller, 2009). However, the inclusion of calciumchannels may have had an impact on the predicted curve for extracellularcurrent injection.

For comparison purposes, we summarized the parameters used in themost relevant theoretical models for equivalent PCs (see Table 1).

2.2.3 Extracellular Current Sources Generated by PCL5. In contrast withprevious studies, we approximated the LFP at any position close to thePCL5 based on the voltage drop in the closest extracellular resistance (seeFigure 4A). LFPs were referenced to a common microelectrode (R) placed atthe most distal compartment of the trunk. In this figure, Re

i represents the ithresistance of the ECS, which is in between the ith and i + 1th compartments,and Ui represents the voltage drop on Re

i . A closeup of the electric circuitfor the ith microelectrode is shown in Figure 4B.

Trivially, the LFP at the ith microelectrode was calculated using the fol-lowing equation (see appendix 4):

Vi =i∑

k=1

Uk.

We used the iCSD method (Pettersen et al., 2006; iCSDplotter software,version 0.1.1) to analyze the distribution of diminutive electric sources (s =−σ∇2ϕ) produced by a particular neuron inside a mesoscopic region (i.e., acortical column) from simulated LFPs (ϕ(�ri) = Vi, i = {1, . . . , N}) at discreterecording sites �ri along the cortical lamina. The parameters used in thisanalysis were the disk diameter d for the sources, which was 0.5 mm; thestandard deviation for the gaussian filter, which was 50 μm; and the electricconductivity, σ (homogeneous media), which was 3 mS/cm (Goto et al.,2010). The thickness (l) of the cortical columns was 2 mm. Assuming thatthe cortical columns were perfect cylinders, their volumes (V = π(d/2)2l)would be 0.39 mm3. We did not use boundary conditions (i.e., free electricpotentials). The mathematical definition of monopoles (m(t)), dipoles (d(t)),and quadrupoles (Q(t)) from the volume current sources (s) are given by

1820 K. Wang, J. Riera, H. Enjieu-Kadji, and R. KawashimaTa

ble

1:C

ompa

riso

nw

ith

Oth

erPr

evio

usT

heor

etic

alM

odel

sfo

rE

quiv

alen

tPC

s.

PCM

odel

s

Mai

nen,

Joer

ges,

Hug

uena

rd,a

ndG

old

etal

.R

all

Des

texh

eTr

aub

etal

.M

urak

amia

ndSe

jnow

ski(

1995

)(2

006)

(196

2)(2

001)

(200

5)O

kad

a(2

006)

Thi

sSt

udy

Sc=

212

Sc=

98Sc

=90

Sc=

10Sc

=11

3Sc

=60

Sc=

NA

Stud

yG

S=

280

GS

=13

6G

S=

284

GS

=15

GS

=18

2G

S=

92G

S=

NA

Tim

eco

nsta

nt(m

s)30

154.

522

4522

.530

Mem

bran

ere

sist

ance

4015

4.5

2250

3040

(K�

-cm

2)

Intr

a/ex

trac

ellu

lar

100/

NA

70/

NA

375/

NA

250/

NA

100/

NA

100/

NA

166/

63re

sist

ance

(�-c

m)

Bra

nchi

ngfo

rmul

aE

quiv

alen

tcyl

ind

erth

eore

mfo

rd

end

riti

ctr

ees

App

end

ix2

Dia

met

er(a

xial

atte

nuat

ion)

Trun

k:Ta

pere

dTa

pere

dC

onst

ant

Piec

ewis

eTa

pere

dC

onst

ant

Tape

red

4.5–

1.4

μm

4–1

μm

15μ

m34

.8μ

m(p

roxi

mal

)4.

0–1.

1μ

m1

μm

4.34

–2.7

6μ

mD

end

rite

s28

.8μ

m(m

idd

le)

0.75

μm

6.2

μm

(dis

tal)

Ele

ctro

toni

cle

ngth

(mm

)0.

60.

11.

40.

7498

0.97

50.

563

0.7

Num

ber

ofco

mpa

rtm

ent

2025

01

361

1010

Ioni

cch

anne

lsPa

ssiv

eN

a-K

-Ca-

IhPa

ssiv

ePa

ssiv

eN

a(F)

-Na(

P)-

Na-

K(D

R)-

K(M

)-N

a-K

f-K

s-Ih

K(D

R)-

K(C

)-K

(C)-

Ca

K(A

)-K

(M)-

K2-

K(A

HP)

-Ca(

L)-

Ca(

T)-

AR

Not

es:I

onic

curr

ents

:Na,

sod

ium

;Na(

F),s

odiu

mfa

st(t

rans

ient

);N

a(P)

,sod

ium

pers

iste

nt;K

,pot

assi

um;K

f,po

tass

ium

fast

;Ks,

pota

ssiu

msl

ow;

K(D

R),

pota

ssiu

mof

del

ayed

rect

ifier

type

;K(C

),po

tass

ium

ofra

pid

volt

age

and

[Ca2

+]i-

dep

end

ent

type

;K(A

),K

(M),

and

K2,

pota

ssiu

mof

tran

sien

tina

ctiv

atin

gA

,M,a

nd2

type

s;K

(AH

P),p

otas

sium

ofsl

ow[C

a2+]

i-d

epen

den

ttyp

e;C

a,ca

lciu

m;C

a(L

),ca

lciu

mof

high

-thr

esho

ldty

pe;

Ca(

T),

calc

ium

oflo

w-t

hres

hold

inac

tiva

ting

type

;Ih,

hype

rpol

ariz

atio

n;A

R,a

nom

alou

sre

ctifi

er.S

c:Sc

opus

ind

ex;G

S:G

oogl

eSc

hola

r.


Intra Extra

iV

1V

NV

iU

1U

NUeNR

eiR

1eR

iUeiR

BA

Figure 4: Modeling the LFP generated by PCL5. (A) Strategy used to simulatethe LFP for our PCL5 model. LFP signals were represented as the electric po-tential differences with respect to a common reference electrode placed nearthe distal trunk in the ECS. The voltage drops in ECS resistances were used torepresent the LFP at each electrode. Re

i stands for the ith resistance of the ECS,which is located between the ith and i + 1th compartment and Ui stands forthe voltage across resistance Re

i . (B) Detailed diagram of the equivalent electriccircuit.

the following equations (Riera et al., 2012):

mz(t)= π

(d2

)2 ∫ l

0s(z, t)dz,

dz(t)= π

(d2

)2 ∫ l

0s(z, t)(z − zm)dz,

Qz(t)= π

(d2

)2 ∫ l

0s(z, t)(z − zm)2dz.


The value zm stands for the laminar coordinate of the center of gravity ofthe cortical column. The axis z is defined in the direction perpendicular tothe neocortex, with positive and negative values toward the supragranularand infragranular layers, respectively.

2.3 Stimulating the PCL5 by Extracellular Current Injection. Becauseof the existence of a resistance in the ECS, we were also able to inject electriccurrents extracellularly to particular compartments of our model, whichallowed us to evoke spiking in the PCL5. In Figure 5, we show the originalelectric circuit that contained the resistance of the ECS (see Figure 5A) andthe modified electric circuit, which accounted for an additional extracellu-lar current source (see Figure 5B). In our model, any injection of an electriccurrent in the ECS close to a compartment of a neuron was representedby an external current source parallel to the extracellular resistance of thatcompartment. Using Thevenin’s theorem, we transformed this parallel cir-cuit into a serial circuit with an equivalent voltage source (see Figure 5C).Hence, we added arbitrary voltage sources to the equivalent electrical cir-cuits for each building block to represent the injected current sources intothe compartments of our model.

The equivalent equation is as follows:

Ein jei = Iin je

i · Rei .

Using Kirchhoff’s laws, we were able to include these serial voltagesources in the original equations for the building blocks (see appendix 2).

3 Results

The resistance of the cellular membranes, ICS, and ECS belonging to someof the building blocks (i.e., apical integrator, trunk, and soma) of our com-partmental models of PCL5 were evaluated based on the mean values oflengths and diameters gathered from 16 PCL5 (see Table 2).

3.1 Intracellular Stimulation of PCL5. Figure 6 (left) shows the prop-agation of an AP train from the soma to the distal trunk in the particularcase of an ECS with zero resistance using our model (see Figure 6A) and theNEURON software (see Figure 6B). We compared single backpropagatingAPs selected from these simulated data with the same initiation times (seeFigures 6A and 6B; close-ups are presented on the right). In the case of Re =0, our model and the NEURON software produced the same results.

Using our model, we were able to estimate the propagation of an APtrain from the soma to the distal trunk in the particular case of an ECSwith nonzero resistance (see Figure 7, left). A single backpropagating APis shown in the panel on the right. In all cases (see Figures 6 and 7), the


Intra ExtraIntra Extra

Intra Extra

eiR

eiR

eiR

injeiE

injeiI

A B

C

Figure 5: A model for extracellular current injection. (A) Original electric circuitcontaining the resistances of the ECS. (B) Extracellular current injection sourcein parallel with the resistance of the ECS. (C) Equivalent circuit in the frequencydomain.

resting membrane potential was held at −70 mV, and a square pulse currentof 200 pA was injected into the soma for 150 ms. The peak amplitude ofAPs remained constant after a very short transitory period. There was goodconcordance between these results and those obtained in a previous exper-iment (Chang & Luebke, 2007). From these simulations, it can be noted that


Table 2: Geometrical Parameters of PCL5.

Length DiameterMEAN ± SD(μm) MEAN ± SD(μm)

IntegratorBranch 1 205.89 ± 25.00 1.26 ± 0.05Last compartment 19.91 ± 1.25 2.48 ± 0.61

PropagatorCompartment 10 79.64 ± 4.99 2.76 ± 0.25Compartment 9 60.47 ± 7.68 2.69 ± 0.55Compartment 8 62.05 ± 5.71 2.80 ± 0.48Compartment 7 61.66 ± 8.69 2.71 ± 0.38Compartment 6 63.29 ± 7.01 2.91 ± 0.47Compartment 5 62.51 ± 7.44 3.01 ± 0.79Compartment 4 61.43 ± 8.69 2.99 ± 0.45Compartment 3 61.93 ± 6.68 3.05 ± 0.45Compartment 2 61.47 ± 5.12 3.33 ± 0.20Compartment 1 60.66 ± 9.06 4.34 ± 0.46

Collector 13.40 ± 1.40

Notes: Statistical analyses were performed using 16 PCL5 with differentlengths and diameters. The mean values and standard deviations (SDs)of the geometrical parameters for all compartments of the integrator,propagator, and collector are shown.

compared with the case of a shunted ECS, the number of spikes createdby our model decreased slightly for Re �= 0. Because of the uncertaintyin determining current leakages in whole-cell voltage clamp experimentsand of the variability in cell dimensions, the number of spikes does notconstitute a suitable experimental criterion to evaluate precisely the impactof extracellular resistance on cellular signaling. We encountered critical nu-merical problems while using the NEURON software for the particular caseof Re �= 0 (data not shown; personal communication with Ted Carnevale,Department of Computer Science and Psychology, Yale University).

To quantify the differences in single APs, we used the amplitude decayand peak delay curves of the backpropagating APs (see Figure 8). First, wechose data from three previous studies reporting the AP decay and delaycurves in PCL5 (Stuart & Sakmann, 1994; Gulledge & Stuart, 2003; Bar-Yehuda et al., 2008; data summarized in Figure 8A). Second, we averagedthe decay and delay values reported in those previous studies at the com-mon sites along the trunk of PCL5. Those three experimental studies werein agreement regarding the decay curves. However, there were notablydifferences between the experimental result of Stuart and Sakmann (1994)and that of the other two studies regarding the estimated delay curves—adiscrepancy of about 3.5 ms at the distal apical dendrites (i.e., 600 μm fromthe soma). Therefore, we compared the decay and delay curves calculated

Modeling Conductivity Profiles of PCs 1825A B

Figu

re6:

Trai

nof

APs

for

aso

mat

icin

trac

ellu

lar

stim

ulat

ion

(our

mod

elve

rsus

NE

UR

ON

).T

hepr

opag

atio

nof

atr

ain

ofA

Psfr

omth

eso

ma

toth

ed

ista

ltru

nkis

show

nfo

rth

eca

ses

ofze

roE

CS

resi

stan

ceus

ing

(A,l

eft)

our

mod

elan

d(B

,lef

t)th

eN

EU

RO

Nso

ftw

are.

We

inje

cted

a20

0pA

curr

ent

puls

ein

toth

eso

ma

from

50to

200

ms.

Inbo

thca

ses,

the

ampl

itud

eof

the

firs

tA

Pw

asla

rger

than

the

ampl

itud

eof

any

subs

eque

ntA

P.In

the

case

ofan

EC

Sw

ith

zero

resi

stan

ce,t

hew

avef

orm

san

dfr

eque

ncy

orra

teof

APs

obta

ined

usin

gou

rm

odel

and

the

NE

UR

ON

soft

war

ew

ere

sim

ilar.

Sing

leA

Pssi

mul

ated

wit

h(A

,rig

ht)o

urm

odel

and

(B,

righ

t)th

eN

EU

RO

Nso

ftw

are

wer

eco

mpa

red

for

the

sam

eti

me

win

dow

.


Figure 7: Effects of ECS resistance on a backpropagating AP. The propagationof a train of APs from the soma to the distal trunk is shown for the cases ofnonzero (left) ECS resistance using our model. As described in Figure 6, weinjected a 200 pA current pulse into the soma from 50 to 200 ms. The amplitudeof the first AP was larger than the amplitude of any subsequent AP. A singleAP (right) simulated within the same time window shown in Figure 6 (right).

from the backpropagating APs in our simulations with those obtained fromaveraging previous experimental data. Finally, we compared the mean ex-perimental curves with those obtained from the simulated backpropagatingAPs (see Figure 8B). We found that the attenuation profile and the velocityof backpropagating APs were strongly affected by the resistance of the ECS,with a better fitting in the case of our model with nonzero resistance for theECS.

When the resistance of the ECS was zero, the theoretical and experimen-tal amplitude decay curves were different (see Figure 8B, left). However, thepredictions obtained from our model using nonzero resistance for the ECSwere very similar to the mean amplitude decay curve estimated experimen-tally. With our model, the mean absolute percentage error (MAPE) was, toa large extent, smaller in the case of nonzero resistance of the ECS (2.7%)compared with the case of zero resistance of the ECS (27.5%). Similarly, forthe peak delay curves, the MAPEs obtained using our model were larger inthe case of having the ECS voltage space clamped (60.2%) compared withthe case of using a highly resistive ECS (38.4%). Further experiments arerequired to verify this conclusion because of the discrepancies in the peakdelay curves estimated experimentally, despite the use of equivalent species(Wistar rats), brain regions (neocortex), and animal ages. Using more con-servative values for the extracellular resistance (e.g., an ECS relative volumeof 6% based on the calculations of Braitenberg and Schutz, 1998), we foundthat the AP decay was stronger and was propagated more slowly towardthe dendritic tuft compared with the case of an ECS/ICS volume fraction of0.30. In such a conservative case, the ECS/ICS volume fraction was equalto 0.064. Therefore, the resistance of the ECS was greater and the effect ofdissipation or retardation more notable.


Figu

re8:

Am

plit

ude

dec

ayan

dpe

akd

elay

curv

esfo

rba

ckpr

opag

atin

gA

Ps.(

A)E

xper

imen

tald

ata

ofd

ecay

and

del

aycu

rves

for

PCL

5A

Ps,w

hich

wer

ed

igit

aliz

edfr

omth

eor

igin

alfi

gure

sre

port

edin

thre

epr

evio

usst

udie

s(S

tuar

t&Sa

kman

n,19

94;G

ulle

dge

&St

uart

,20

03;

Bar

-Yeh

uda

etal

.,20

08).

We

aver

aged

the

dat

ato

obta

inth

eva

lues

wit

hth

ebe

stlik

elih

ood

for

the

dec

ayan

dd

elay

curv

es(c

onti

nuou

slin

es).

Toob

tain

valu

esfo

rsi

tes

dis

tant

from

the

som

a,w

eex

tend

edth

atav

erag

edd

ata

usin

ga

linea

lin

terp

olat

ion

met

hod

(das

hed

lines

).(B

)C

ompa

riso

nof

the

dec

ayan

dd

elay

curv

esfo

rth

esi

mul

ated

APs

wit

hth

eav

erag

edex

peri

men

tald

ata.

For

the

case

ofR

e�=

0,an

EC

S/IC

Svo

lum

efr

acti

onof

0.3

was

used

toes

tim

ate

the

extr

acel

lula

rre

sist

ance

s.T

hecu

rves

corr

espo

ndin

gto

Bra

iten

berg

and

Schu

tz(1

998)

wer

eob

tain

edus

ing

our

mod

elan

dan

EC

S/IC

Svo

lum

efr

acti

onof

0.06

4.


Using the strategy proposed in the section 2, we simulated the LFP usingour model and calculated the respective CSD in the case of an ECS withnonzero resistance. An electric current pulse with an amplitude of 200 pAand a duration of 5 ms was injected into the intracellular space to achievethe firing of a single AP from PCL5 (see Figure 9). The CSD spatiotempo-ral pattern calculated was similar to that estimated previously based onexperimental data (Swadlow, Gusev, & Bezdudnaya, 2002; Bereshpolovaet al., 2007). Based on our calculated CSD, we determined the multipolarcomponents generated by a single backpropagating AP along the trunk ofa PCL5. Note that our estimators are valid only if the microelectrodes arein close proximity to the PCL5 (e.g., a linear probe with microelectrodesarranged in parallel to the PCL5 at a distance of less than 50 μm from it).In contrast to the results obtained by Milstein and Koch (2008), we wereable to distinguish monopolar and quadrupolar contributions to the LFP,which were comparable in size to those of the dipolar source model. Thestrategy that Milstein and Koch (2008) described consisted of first solvingthe discrete cable equation assuming a zero resistance for the ECS and thenusing the resulting transmembrane currents (i.e., both primary and return-ing currents) to calculate the LFP using the Poisson equation. Therefore,their results originated from both the local dipolar character of the trans-membrane current sources and the instantaneous propagation of the electricfield in pure resistive media (i.e., the quasi-static approach). In our case, thestrategy used to calculate the LFP took into account dynamic changes inthe extracellular potentials that are influenced by the existence of capacitiveand resistive elements in the cellular membranes.

3.2 Extracellular Stimulation. We used the proposed strategy to stim-ulate PCL5 using extracellular current injection at the level of each com-partment. We applied extracellular electric currents (square pulses of 50–200 ms) at different positions of the PCL5 to generate a train of APs in itssoma. The amplitudes of these electric currents were adjusted to reproducetrains of AP with similar spiking rate. Figure 10 illustrates this procedurefor four positions: the soma and sites in the proximal, middle, and distaltrunk.

Figure 11 depicts the relationship between the amplitudes at which theelectric currents injected into the ECS generate a similar spike train and thedistance from the soma. Trivially, the microelectrodes located farther fromthe soma required a higher current to stimulate the PCL5 equivalently. Thedistance from the soma and the amplitude of the current injected exhibitedan exponential-like relationship. This kind of sensibility profile of the PCL5to an extracellular current injection will allow us to create, in the near future,strategies to stimulate this cell type selectively using specially designedMEAs and current-injection protocols.


Figu

re9:

LFP

and

CSD

anal

ysis

.The

back

prop

agat

ing

mem

bran

epo

tent

ials

thro

ugh

allc

ompa

rtm

ents

ofth

epr

opag

ator

ofth

ePC

L5

are

show

non

the

top-

left

pane

l.T

hepa

rtic

ular

inje

ctio

npr

otoc

ol(s

quar

epu

lse

curr

ento

f200

pAin

toth

eso

ma

for

ad

urat

ion

of5

ms)

caus

edth

isne

uron

type

toge

nera

tea

sing

leA

P.L

FPs

wer

ege

nera

ted

bysu

cha

back

prop

agat

ing

AP

(top

-rig

htpa

nel)

.A

colo

rpa

nel

show

sth

esp

atio

tem

pora

lC

SDpa

tter

n(b

otto

m-l

eft

pane

l),w

hich

was

calc

ulat

edfr

omth

eL

FPs

usin

gth

eiC

SDm

etho

d.T

his

patt

ern

was

very

sim

ilar

toth

ose

obse

rved

expe

rim

enta

llyby

Ber

eshp

olov

aet

al.(

2007

).T

heti

me

seri

esw

ith

the

mul

tipo

lar

mom

ents

(i.e

.,m

onop

oles

,dip

oles

,and

quad

rupo

les)

are

show

non

the

bott

om-r

ight

pane

l.


Figure 10: Membrane potentials caused by current injection in the ECS. Fourpositions (soma and proximal/middle/distal trunk) were chosen to illustratethe neuronal response to current injection in the ECS. The amplitude of theinjected current was adjusted at all four locations to keep a similar firingrate.


Figure 11: Spatial profile of the current injection. The exponential-like relation-ship between the amplitudes required for the current injection to produce asimilar output pattern in this type of neuron along the PCL5 trunk is shown.

4 Discussion

Riera et al. (2006) described the electrotonic propagation of the membranepotential in PCL5 in the human visual cortex using a three-compartmentmodel that included the apical dendrites, basal dendrites, and the soma.These authors used this model to estimate crucial physiological parametersin the cortical microcircuit from large-scale EEG data that were obtainedfrom healthy subjects undergoing a flickering checkerboard visual stimu-lation paradigm (Riera et al., 2007). In the current study, we extended thisprevious model not only to include active ion currents but also to representcells with other morphologies via the combination of four basic electrotonicdevices as basic building blocks: the integrator, the propagator, the 3D con-nector, and the collector. Each of these devices, which in principle couldcomprise multiple compartments, is able to connect to any other device bymeans of terminals (open circuits) with free voltage differences as the link-ing physical magnitudes. As in Riera et al. (2006), the extracellular electricpotential was defined as the voltage drops in the resistances of the extracel-lular space for each compartment. Additionally, we obtained mathematicalformulas for the case in which exogenous sources of electric currents werein parallel to these extracellular resistances. The theoretical result of thelatter analysis allowed us to create a realistic profile of PCL5 sensitivity toexternal current stimulation by MEAs located in close proximity to thesecells. As a result of the increased use of prosthetic devices to stimulate


directly the neocortex via the injection of electric currents into the ECS, thesubjects discussed in this study are of particular relevance for the futuredevelopment of brain-machine interfaces.

Moreover, the proposed model provides a link between the intracellular(membrane potentials) and extracellular (LFP/MU(S)A) recordings withoutusing a quasi-static approach for the electric fields. However, our theoreticalmodel is valid only for recording sites located in close proximity to theneurons of interest. We were able to reproduce spatiotemporal patterns inthe extracellular CSD generated by backpropagating APs that were similarto those reported experimentally by Swadlow et al. (2002) and Bereshpolovaet al. (2007). When the resistance of the ECS was different from zero, ourmodel was able to reproduce the observed curves for the amplitude decayand peak delay of backpropagating APs in PCL5. Based on the resultsobtained in this study, we conclude that including a nonzero resistance inthe ECS is crucial for the evaluation (using theoretical simulations) of therole of the apical dendrites of the PCL5 in the integration of informationalong the cortical layers. We demonstrated that the membrane potentialssuffer more attenuation during propagation back from the soma to theapical dendrites in situations where the resistances for the ECS are large.Because of the existence of the hyperpolarization-activated cation current,which shows larger conductivity values toward the apical dendrites, theeffect of a nonzero resistance of the ECS on the spatiotemporal integrationof postsynaptic potentials must be evaluated in future studies.

4.1 The Decay and Delay for Backpropagating APs: The Discrepan-cies. To evaluate the impact of introducing an extracellular resistance inthe multicompartmental models of PCL5, we used the decay and delaycurves of backpropagating APs obtained in these particular cell types bydifferent laboratories (Stuart & Sakmann, 1994; Gulledge & Stuart, 2003; Bar-Yehuda et al., 2008). The first report dates from the early 1990s when Stuartand Sakmann (1994) performed patch clamp recordings from dendrites ofneocortical pyramidal cells using parasagittal neocortical brain slices from2-week-old Wistar rats. Almost 10 years later, Gulledge and Stuart (2003)obtained similar measurements from coronal brain slices containing theprelimbic prefrontal cortex from 3- to 5-week-old Wistar rats. Stuart andSakmann (1994) described a decay curve that was similar to that reportedby Gulledge and Stuart (2003). The delays, however, were much larger: theAPs reached a site about 600 μm from the soma and 3.5 ms slower comparedto the results of Gulledge and Stuart (2003). More recently, Bar-Yehuda et al.(2008) used sagittal slices from 5- to 7-week-old Wistar rats to obtain decayand delay curves that were similar to those reported by Gulledge and Stuart(2003). We realized that there is a discrepancy between the delay curve re-ported in the initial work by Stuart and Sakmann (1994) and those obtainedin more contemporary studies (Gulledge & Stuart, 2003; Bar-Yehuda et al.,2008). These three studies used Wistar rats. We opted to average the data


from these three studies for both decay and delay curves. We believe thatadditional studies are required to verify whether these curves are affectedby age, species, brain region, or PCL5 subtype. Currently, we believe thatthe averaged data provide the most credible curves.

As shown in Figure 8, a multicompartment model for PCL5 that includesextracellular resistances generated APs that propagated more slowly to-ward the apical tuft. To explain this phenomenon, we used a simplifiedmodel of a membrane circuit (i.e., one single compartment) that includednot only membrane and intracellular resistances but also a nonzero extra-cellular resistance. The membrane potential can be calculated using

CdVdt

= − 1R∗ V + Iext,

in which R∗ is an equivalent resistance that results from having a mem-brane resistance in parallel to a series of intracellular Ri and extracellular Reresistances. The time constant for the circuit is τ = R∗C. Therefore,

1R∗ = 1

Rm+ 1

Rei= Rm + Rei

RmRei,

where Rei = Re + Ri.As mentioned earlier, Re = 1.3Ri. We compared the R∗ in the cases “Re is

not zero” and “Re is zero”:

1R∗

Re �=0= Rm + Rei

RmRei= Rm + 2.3Ri

2.3RiRm,

1R∗

Re=0= Rm + Ri

RmRi= Rm + Ri

RiRm.

It can be easily demonstrated thatR∗

Re �=0

R∗Re=0

> 1, which implies τRe �=0 > τRe=0.

An alternative explanation for the AP retardation in the case of nonzeroextracellular resistive originates from a thermodynamic viewpoint. Thenet production of heat in a given time window Q = IRt will be higherfor electrical circuits with more resistive elements, which are dissipative.Such an energy transformation process might underlie larger delays in thepropagation of APs.

4.2 The Charge-Balanced Hypothesis: Extracellular Potential Models.In a classical approach, the connection between intracellular and extracel-lular recordings is based on the assumption of the existence of microscopiccurrent sources across the cellular membrane of whole neurons. Therefore,


the extracellular electric potentials are naturally calculated using the Pois-son equation for quasi-static electric fields. This strategy might be valid anduseful in situations where the microelectrodes used to record the extracel-lular potentials are located far from the neuronal populations of interest.However, the classic approach may produce inaccurate results in situationswhere these microelectrodes are in close proximity to the neurons that gen-erate the extracellular potentials. In addition, such a source model impliesa microscopic charge balance in the cellular membranes. Riera et al. (2012)provided recent evidence that refutes such a working hypothesis at themesoscopic level. In the barrel cortex of Wistar rats undergoing a whiskerstimulation protocol, they found important contributions from monopo-lar and quadrupolar current sources to the extracellular potentials at themesoscopic level. The role of multipolar current sources in the genesis ofLFP/MU(S)A has been evaluated (Milstein & Koch, 2008): dipolar compo-nents were suggested to be larger than other multipolar moments. There-fore, further evaluations are needed to clarify whether the dipolar momentis the major component of LFPs.

4.3 Poisson Equation versus Kirchhoff’s Circuit Laws. The majorproblem of previous strategies that attempted to link models for the intra-cellular and extracellular experimental approaches is the strong inconsis-tency among the underlying assumptions. Researchers use the quasi-staticapproach for the electric field, which was initially introduced to describethe propagation of electric and magnetic fields inside biological tissues(Plonsey & Heppner, 1967) to estimate the extracellular electric potentialϕ(�r) everywhere from known microscopic current sources (s) across themembrane of neurons. For the electric field, this approach leads to thewell-known Poisson equation (∇ · (σ∇ϕ) = −s). In the frequency range ofthe electrophysiological phenomena (ω � 100 kHz), dispersive effects inthe tissues are ignored (i.e., pure resistive media). Under such a condition,the conductivity (σ ) reflects mostly the macroscopic conductivity, which isdetermined by the characteristics of the cell suspension (i.e., ECS/ICS vol-ume fraction) and does not include any contribution originating from theexistence of highly dispersive membranes. Currently, we are familiar withthe fact that such an approach is not valid even for a lower LFP frequencyrange (ω � 100 Hz) (Gabriel, Gabriel, & Corthout, 1996; Gabriel, Lau, &Gabriel, 1996a, 1996b; Gabriel, Peyman, & Grant, 2009; Bedard & Destexhe,2009; Bedard, Rodrigues, Roy, Contreras, & Destexhe, 2010). The differentialequations used to describe the membrane potentials in neurons originate bynature from the fact that there is a capacitor separating the intracellular andextracellular spaces. As demonstrated in this letter, we have to be carefulwhen representing these actual current sources at a mesoscopic level usingequivalent dipolar models that neglect the contribution of monopolar andhigh-order multipolar moments to the LFPs (Riera et al., 2012).


Appendix 1: Calculation of Resistances

In every compartment, l is the length, d is the diameter, A is the area, and V isthe volume. The parameters A and V of every compartment were calculatedfrom the l and d values. The calculation of the membrane resistance (Rm) isgiven by:

Rm = rm

1.92 × A.

Here, the membrane resistivity (rm) is equal to 40,000 (� × cm2). A factorof 1.92 was introduced to account for the area of dendritic spines (Rhodes,2006).

The calculation of the resistance of the ICS (Ri) is given by

Ri = ri × lAi

.

Here, the resistivity of the ICS (ri) is equal to 166 (� × cm).The calculation of the resistance of the ECS (Re) is given by

Re = 1.3 × Ri.

Proof.

Re

Ri=

re×lAe

ri×lAi

= re

ri× Ai × l

Ae × l= re

ri× Vi

Ve= 63

166× 1

0.3∼.= 1.3.

Here, the resistivity of the ECS (re) is equal to 63 (� × cm). We used thefact that the ECS/ICS volume fraction is approximately 0.3.

For the simulations performed using the NEURON software, the axialextracellular resistance was defined as xraxial = re/(0.3 × Ai) (dimension −M�/cm).

Appendix 2: Model of Building Blocks

The membrane unit of a cell can be modeled as an RC circuit and a primarycurrent source, all in parallel (see Figure 12A). R and C stand for the mem-brane resistance and capacitance, respectively. In the frequency domain, anequivalent complex resistance of the circuit is defined as R∗ = R

1+ jωτ, where

τ = RC represents the membrane time constant.


Figure 12: The RC circuit and branching rules for dendrites. (A) The originalelectric circuit (left) for the cell membrane is shown on the left. It is constitutedby an RC circuit in parallel with a membrane current source. The symbolsRm and C stand for the membrane resistance and capacitance, respectively. Inthe frequency domain configuration, an equivalent complex resistance of thecircuit is defined as R∗

m = Rm1+ jωτ

, where τ = RmC represents the membrane timeconstant. In the frequency domain, such an electric circuit is equivalent to aparallel circuit with a current source and a complex resistance (middle). Usingthe Thevenin equivalent theorem, the parallel circuit with a current source anda resistance can be transformed into a serial circuit with an electromotive forceand a resistance (right). (B) Branching rules for oblique (left) and basal (right)dendrites.

A. Model of the Integrator. The integrator (see Figures 1A.2 and 1B.2)represents a device that possesses many branches with heterogeneousimpedances, receives a huge number of inputs, and produces a single out-put (i.e., a voltage difference). The integrator is useful to represent dendriticbranches. This output is used as the input to another electrotonic device.


According to the actual morphometry of dendritic trees (Larkman, 1991a,1991b, 1991c), the integrator might comprise intermediate and terminalbranches. We considered a model of the integrator composed mainly of aset of intermediate branches.

To determine the dynamic equations to calculate the membrane poten-tials in each branch of the integrator, as well as in its last compartment,we used Kirchhoff’s laws. The equivalent circuit for this electrotonic de-vice is shown in Figure 2A, and the following equations were obtainedstraightforwardly:

−V0 − ReiTI − Re

TIin jeT − Rei

A1IA1

− ReA1

Iin jeA1

+ VA1= 0 (A:a1)


TIin jeT − Rei

A2IA2

− ReA2

Iin jeA2

+ VA2= 0 (A:a2)

. . . . .

. . . . .

. . . . .


TIin jeT − Rei

AkIAk

− ReAk

Iin jeAk

+ VAk= 0 (A:ak)

. . . . .

. . . . .

. . . . .

− V0 − ReiTI − Re

TIin jeT − Rei

AMIAM

− ReAM

Iin jeAM

+ VAM= 0 (A:aM)

I =M∑

k=1

IAk(A:aM+1)

−V0 + R∗TIO + E∗

T = 0 (A:aM+2)

−V1 − ReTIe

1 − ReTIin je

T − RiTIi

1 + R∗TIO + E∗

T = 0 (A:aM+3)

I = IO + Ie1 = IO + Ii

1 = 0, (A:aM+4)

where Reiχ = Re

χ + Riχ , χ = {T, Ak}.

The superscript e and i denote the extracellular and intracellular spaces.IAk

stands for a current entering the kth branch of the integrator. Iin jeAk

standsfor an extracellular injection current at the extracellular resistance of the kthbranch of the integrator. Iin je

T stands for an extracellular injection current atthe extracellular resistance of the last compartment of the integrator.


The expression of current I is obtained by dividing each A:akth equationby Rei

Akand summing all of them, as follows:

I =∑M

k=1

VAkRei

Ak

− V0∑M

k=11

ReiAk

−Iin jeT Re

T

∑Mk=1

1Rei

Ak

− ∑Mk=1

Iin jeAk

ReAk

ReiAk

1 + ∑Mk=1

ReiT

ReiAk

. (A:b1)

Plugging equation A:b1 into the A:akth equation enabled us to determinethe electric current flowing into a kth branch of the integrator as follows:

IAk= 1

ReiAk

⎡⎢⎣VAk

− V0 −∑M

k=1Rei

TRei

Ak

VAk− V0

∑Mk=1

ReiT

ReiAk

1 + ∑Mk=1

ReiT

ReiAk

−

⎛⎜⎝1 −

∑Mk=1

ReiT

ReiAk

1 + ∑Mk=1

ReiT

ReiAk

⎞⎟⎠ Re

TIin jeT +

∑Mk=1

ReiT

ReiAk

ReAk

Iin jeAk

1 + ∑Mk=1

ReiT

ReiAk

− ReAk

Iin jeAk

⎤⎥⎦ .

(A:b2)

Hence, coupling equation A:b2 with the following equations,

VAk= E∗

Ak− R∗

AkIAk

, (A:b3)

E∗Ak

= RAkIIAk

, (A:b4)

allowed us to obtain an equation that describes the change in the membranepotential in each of the branches of the integrator:

dVAk

dt= 1

τm

⎡⎢⎣RAk

IIAk

−(

1 +RAk

ReiAk

)VAk

+

⎛⎜⎝

∑Ml=1

ReiT

ReiAl

VAl

1 + ∑Ml=1

ReiT

ReiAl

⎞⎟⎠ RAk

ReiAk

+

⎛⎜⎝ 1

1 + ∑Ml=1

ReiT

ReiAl

⎞⎟⎠ RAk

ReiAk

V0 + VEXTRA

⎤⎥⎦ , (A:b5)

VEXTRA =RAk

ReiAk

⎡⎢⎣

⎛⎜⎝1 −

∑Ml=1

ReiT

ReiAl

1 + ∑Ml=1

ReiT

ReiAl

⎞⎟⎠ Re

TIin jeT

+

⎛⎜⎝Re

AkIin jeAk

−∑M

l=1Rei

TRei

Al

ReAl

Iin jeAl

1 + ∑Ml=1

ReiT

ReiAl

⎞⎟⎠

⎤⎥⎦ . (A:b6)


The following expressions of the electric currents were obtained basedon equations A:aM+2 and A:aM+4:

IO1 = V0 − E∗

T

R∗T

, (A:b7)

Ie1 = Ii

1 = V0 − V1

ReiT

. (A:b8)

The equation describing the dynamics of the membrane potential in thelast compartment of the integrator was obtained by inserting equations(A:b7) and (A:b8) into equation (A:aM+3). Thus, we obtained:

dV0

dt= 1

τm

⎡⎢⎢⎣RTII

0 +RT

∑Ml=1

VAlRei

Al

1 + ∑Ml=1

ReiT

ReiAl

−

⎛⎜⎝1 + RT

ReiT

+RT

∑Ml=1

1Rei

Al

1 + ∑Ml=1

ReiT

ReiAl

⎞⎟⎠V0

+RT

ReiT

V1 + VEXTRA

⎤⎥⎥⎦ , (A:b9)

VEXTRA =−

⎛⎜⎝

∑Ml=1

RTRei

Al

1 + ∑Ml=1

ReiT

ReiAl

⎞⎟⎠ Re

TIin jeT −

∑Ml=1

RTRei

Al

ReAl

Iin jeAl

1 + ∑Ml=1

ReiT

ReiAl

+ RT

ReiT

ReTIin je

T .

(A:b10)

B. Model of the Propagator. The propagator (see Figure 1A.2) refers toan electrotonic device along which an AP can propagate. As the morphom-etry is nonuniform within a compartment of the propagator, we assumedthat all compartments can be divided into two subcompartments (upperand lower parts). Thus, the following equations were obtained by apply-ing Kirchhoff’s voltage law to all meshes of the propagator’s equivalentelectrical circuit (see Figure 2.B):

−V0 + VT1+ RUe

T1IUeT1

+ RUeT1

Iin jeT1

+ RUiT1

IUiT1

= 0 (B:a1)

−VT1+ VT2

+ RLeT1

ILeT1

+ RLiT1

ILiT1

+ RUeT2

IUeT2

+ RUeT2

Iin jeT2

+ RUiT2

IUiT2

= 0 (B:a2)

−VT2+ VT3

+ RLeT2

ILeT2

+ RLiT2

ILiT2

+ RUeT3

IUeT3

+ RUeT3

Iin jeT3

+ RUiT3

IUiT3

= 0 (B:a3)

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .


−VTk−1+ VTk

+ RLeTk−1

ILeTk−1

+ RLiTk−1

ILiTk−1

+ RUeTk

IUeTk

+ RUeTk

Iin jeTk

+ RUiTk

IUiTk

= 0

(B:ak)

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

− VTN−1+VTN

+RLeTN−1

ILeTN−1

+RLiTN−1

ILiTN−1

+RUeTN

IUeTN

+RUeTN

Iin jeTN

+RUiTN

IUiTN

= 0

(B:aN-1)

− VTN+ VTN+1

+ RLeTN

ILeTN

+ RLeTN

Iin jeTN

+ RLiTN

ILiTN

= 0 (B:aN)

By applying Kirchhoff’s current law, the following node equations (seeequations B:a1–B:aN) are obtained:

ILiTk

= IUiTk+1

, (B:aN+1)

ILeTk

= IUeTk+1

, (B:aN+2)

IUeT1

= IUiT1

, (B:aN+3)

IUeT2

= IUiT2

, (B:aN+4)

IUiT2

= IUeT2

= IUiT1

− IT1, (B:aN+5)

IUiT3

= IUeT3

= IUiT1

− (IT1+ IT2

), (B:aN+6)

IUiT4

= IUeT4

= IUiT1

− (IT1+ IT2

+ IT3), (B:aN+7)

. . . . .

. . . . .

. . . . .

IUeTk

= IUiTk

= IUiT1

−N−1∑p=1

ITp, (B:aN+8)

IUeTN+1

= IUiTN+1

= IT1−

N−1∑p=1

ITp− ITN

, (B:aN+9)


can be rewritten as follows:

−(V0 − VT1) + RUei

T1IUiT1

+ RUeT1

Iin jeT1

= 0, (B:b1)

−(VTk−1− VTk

) + (RTk−1+ RUei

Tk)IUi

Tk+ RUe

TkIin jeTk

= 0, (B:b2)

−(VTN− VTN+1

) + RTNIUiTN

− RLeiTN

ITN+ RLe

TNIin jeTN

= 0, (B:b3)

where RXeiTk

= RXeTk

+ RXiTk

, X = {L,U}.The superscript e and i denote the extracellular and intracellular spaces.

The superscript U and L denote the upper part and lower part of every com-partment of the propagator. Iin je

Tkstands for an extracellular injection current

at the extracellular resistance of the Tth compartment of the propagator.Based on the equations presented above, the expressions of the currentsflowing into particular resistances of the propagator are given as follows:

IUiT1

=V0 − VT1

RUeiT1

−RUe

T1

RUeiT1

Iin jeT1

, (B:b4)

IT1=

V0 − VT1

RUeiT1

+VT2

− VT1

RLeiT1

+ RUeiT2

+RUe

T2

RLeiT1

+ RUeiT2

Iin jeT2

−RUe

T1

RUeiT1

Iin jeT1

, (B:b5)

ITk=

VTk−1− VTk

RLeiTk−1

+ RUeiTk

+VTk+1

− VTk

RLeiTk

+ RUeiTk+1

+RUe

Tk+1

RLeiTk

+ RUeiTk+1

Iin jeTk+1

−RUe

Tk

RLeiTk−1

+ RUeiTk

Iin jeTk

(B:b6)

2 � k � N − 1

ITN=

VTN−1−VTN

RLeiTN−1

+RUeiTN

+VTN+1

−VTN

RLeiTN

+RLe

TN

RLeiTN

Iin jeTN+1

−RUe

TN

RLeiTN−1

+RUeiTN

Iin jeTN

, (B:b7)

ILiTN

=VTN

− VTN+1

RLeiTN

−RLe

TN

RLeiTN

Iin jeTN+1

. (B:b8)

The expression of electric voltage through the membrane in the kth com-partment is defined as

VTk= E∗

Tk+ R∗

TkITk

. (B:b9)

After some algebraic manipulations, equations B:b4 to B:b9 enabled us tofind the final expressions for the membrane potentials at all compartments


of the propagator:

dVT1

dt= 1

τm

[RT1

IIT1

+RT1

RUeiT1

V0 −(

1 +RT1

RUeiT1

+RT1

RLeiT1

+ RUeiT2

)VT1

+RT1

RLeiT1

+ RUeiT2

VT2+ VEXTRA1

], (B:b10)

VEXTRA1= RT1

(RUe

T2

RLeiT1

+ RUeiT2

Iin jeT2

−RUe

T1

RUeiT1

Iin jeT1

), (B:b11)

dVTk

dt= 1

τm

[RTk

IITk

+RTk

RLeiTk−1

+ RUeiTk

VTk−1

−(

1 +RTk

RLeiTk−1

+ RUeiTk

+RTk

RLeiTk

+ RUeiTk+1

)VTk

+RTk

RLeiTk

+ RUeiTk+1

VTk+1

+ VEXTRAk

], (B:b12)

VEXTRAk= RTk

(RUe

Tk+1

RLeiTk

+ RUeiTk+1

Iin jeTk+1

−RUe

Tk

RLeiTk−1

+ RUeiTk

Iin jeTk

), (B:b13)

2 � k � N − 1

dVTN

dt= 1

τm

[RTN

IITN

+RTN

RLeiTN−1

+ RUeiTN

VTN−1

−(

1 +RTN

RLeiTN−1

+ RUeiTN

+RTN

RLeiTN

)VTN

+RTN

RLeiTN

VTN+1+ VEXTRAN

], (B:b14)

VEXTRAN= RTN

(RLe

TN

RLeiTN

Iin jeTN+1

−RUe

TN

RLeiTN−1

+ RUeiTN

Iin jeTN

). (B:b15)

C. Model of the 3D Connector. The 3D connector (see Figure 1A.2)refers to an electrotonic device that links three particular building blocks tosimulate a point of branch bifurcations. Every branch of the 3D connectorreceives and generates both input and output electric potentials. Withoutlosing generality, we assumed that the 3D connector contains two inputs


and one single output. The application of Kirchhoff’s laws within all meshesof the equivalent electrical circuit of the 3D connector (see Figure 2C) led tothe following equations:

−VI1 + Rei

1 II1 + Re

1Iin je1 + VM = 0, (C:a1)

−VI2 + Rei

2 II2 + Re

2Iin je2 + VM = 0, (C:a2)

−VO + ReiOII

O + ReOIin je

O + VM = 0, (C:a3)

IM = II1 + II

2 + IIO, (C:a4)

−VM + R∗MIM + E∗

M = 0, (C:a5)

E∗M = R∗

MIIM, (C:a6)

where Reiχ = Re

χ + Riχ , χ = {0, 1, 2}.

The superscript e and i stand for the extracellular and intracellular spaces.IIχ stands for a current entering the three branches of the 3D connector. Iin je

χ

stands for an extracellular injection current at the extracellular resistanceof the three branches of the 3D connector. Based on the equations pre-sented above, the expressions of currents flowing into the extracellular andintracellular resistances of the 3D connector are given as follows:

II1 = VI

1 − VM − Re1Iin je

1

Rei1

, (C:a7)

II2 = VI

2 − VM − Re2Iin je

2

Rei2

, (C:a8)

IIO = VO − VM − Re

OIin jeO

ReiO

. (C:a9)

Based on the equations presented above, we obtained an equation thatdescribes the changes in the membrane potential in the 3D connector:

dVM

dt= 1

τm

[RMII

M + RM

Rei1

VI1 + RM

Rei2

VI2 + RM

ReiO

VO

−(

1 + RM

Rei1

+ RM

Rei2

+ RM

ReiO

)VM + VEXTRA

], (C:b1)

VEXTRA =−RM

Rei1

Re1Iin je

1 − RM

Rei2

Re2Iin je

2 − RM

ReiO

ReOIin je

O . (C:b2)


D. Model of the Collector. The collector (see Figures 1A.2 and 1B.2)represents an electrotonic device that collects many inputs coming fromother devices. Based on its equivalent electrical circuit, which is representedin Figure 2D, we obtained the following equations:

VS − ReiS I − Re

SIin jeS − Rei

S1IS1

− ReS1

Iin jeS1

+ VS1= 0, (D:a1)


SIin jeS − Rei

S2IS2

− ReS2

Iin jeS2

+ VS2= 0, (D:a2)

. . . . .

. . . . .

. . . . .


SIin jeS − Rei

SkISk

− ReSk

Iin jeSk

+ VSk= 0, (D:ak)

. . . . .

. . . . .

. . . . .


SIin jeS − Rei

SMISM

− ReSM

Iin jeSM

+ VSM= 0, (D:aM)

I =M∑

k=1

ISk, (D:aM+1)

VS − E∗S − R∗

SI = 0, (D:aM+2)

VSk+ R∗

SkIk − E∗

Sk= 0, (D:aM+3)

VSk+ Re

TkIek + Re

TkIin jek + Ri

TkIik − VTk

= 0, (D:aM+4)

ISk= Ik + Ie

k = Ik + Iik, (D:aM+5)

where ReiS = Re

S + RiS.

ISkrepresents currents that flow into a branch of the collector. The super-

script e and i denote the extracellular and intracellular spaces. Iin jeSk

standsfor an extracellular injection current at the extracellular resistance of the kthbranch of the collector. Iin je

S stands for an extracellular injection current atthe extracellular resistance of the soma compartment of the collector. Af-ter dividing each kth equation by Rei

Skand summing up all equations, the

following expression of the electric current was obtained:

I =∑M

k=1

VSkRei

Sk

− VS∑M

k=11

ReiSk

− Iin jeS Re

S

∑Mk=1

1Rei

Sk

− ∑Mk=1

Iin jeSk

ReSk

ReiSk

1 + ReiS

∑Mk=1

1Rei

Sk

. (D:b1)


Based on equations D:aM+2 and D:b1, we found that the dynamics ofthe membrane potential in the collector are governed by

dVS

dt= 1

τm

⎡⎢⎢⎣RSII

S +RS

∑Ml=1

VSlRei

Sl

1 + ∑Ml=1

ReiS

ReiSl

−

⎛⎜⎝1+

RS∑M

l=11

ReiSl

1+ ∑Ml=1

ReiS

ReiSl

⎞⎟⎠VS +VEXTRA

⎤⎥⎥⎦,

(D:b2)

VEXTRA = − RS

ReiS

Iin jeS Re

S

∑Ml=1

1Rei

Sl

1Rei

S+ ∑M

l=11

ReiSl

− RS

ReiS

∑Ml=1

Iin jeSk

ReSl

ReiSl

1Rei

S+ ∑M

l=11

ReiSl

. (D:b3)

The following expressions of currents were obtained from equationsD:aM+2, D:aM+3, D:aM+4, D:aM+5, and D:ak:

ISk=

E∗S − VSk

R∗S

+VTk

− VSk− Re

TkIin jek

ReiTk

. (D:b4)

After inserting equation D:b4 into the D:akth equation and some alge-braic calculations, we found that the dynamics of the membrane potentialat the kth entrance of the collector is governed by the following equation:

dVSk

dt= 1

τm

[RSk

IISk

+RSk

ReiTk

VTk−

(1 +

RSk

ReiTk

+RSk

ReiSk

)VSk

+

⎛⎜⎝

∑Ml=1

ReiS

ReiSl

VSl

1+ ∑Ml=1

ReiS

ReiSl

⎞⎟⎠RSk

ReiSk

+

⎛⎜⎝ RSk

1+ ∑Ml=1

ReiS

ReiSl

⎞⎟⎠ VS

ReiSk

+ VEXTRA

⎤⎥⎦ ,

(D:b5)

VEXTRA =−RSk

ReTk

ReiTk

Iin jek −

RSk

ReiSk

Iin jeS Re

S

∑Ml=1

1Rei

Sl

1Rei

S+ ∑M

l=11

ReiSl

−RSk

ReiSk

∑Ml=1

Iin jeSk

ReSl

ReiSl

1Rei

S+ ∑M

l=11

ReiSl

.

(D:b6)

Here, IISk

= ISSk

+ IiSk

represents the sum of the synaptic and ionic currents.


Appendix 3: Electrotonic Parameters of Oblique and Basal Dendrites

We estimated equivalent input impedances for the oblique and basal den-drites by using recent anatomical data from the work of Romand et al.(2011).

A. Model of the Oblique Dendrites. The oblique dendrites representa device that contains 15 intermediate and 24 terminal segments whosenumbers, lengths, and diameters in P14 rats were reported by Romandet al. (2011; see Figure 3, left C1). Taking into account this information, weformulated a branching rule for this particular dendritic tree, as illustratedin Figure 12B, left. In this case, the final differential equation for the branchof the 3D connector that is linked to the oblique dendrites is given by

dVO

dt= 1

τm

[Rm

OBIIOB + Rm

OB

RieOB

VM −(

RmOB

RieOB

+ 1)

VO]

.

For details regarding how the voltage difference VO relates to the mem-brane potential in the 3D connector VM, we refer readers to equation C:b1in appendix 2, section C. The equivalent resistances Rm

OB and RieOB are given

by the following general equation, with k = {m, ie}.

RkOB = Rk

OBi +Rk

OBi +Rk

OBi +Rk

OBi + RkOBt

32

22

Here, RkOBt is the kth resistance of the terminal segments, and Rk

OBi is thekth resistance of intermediate segments, both in oblique dendrites. Theseresistances can be estimated using the geometrical parameters reported byRomand et al. (2011) in the formulas developed in appendix 1. Note that forany segment type, l = {i, t} Rie

OBl = RiOBl + Re

OBl ; that is, the resistance, RieOBl ,

is calculated as the sum of the intracellular and extracellular longitudinalresistances of that segment.

B. Model of the Basal Dendrites. The basal dendrites represent a de-vice that contains 13 intermediate and 24 terminal segments whose num-bers, lengths, and diameters in P14 rats were reported by Romand et al.(2011; see Figure 3, left C2). Taking into account this information, we for-mulated a branching rule for this particular dendritic tree, as illustrated inFigure 12.B (right). In this case, the final differential equation for the basal


input to the collector is given by

dVS2

dt= 1

τm

⎡⎢⎣RS2

IIS2

+RS2

RieBA

VBA −(

1 +RS2

RieBA

+RS2

RieS2

)VS2

+ VS

RieS2

+ 1Rie

S2

⎛⎜⎝

∑2l=1

1Rie

Sl

VSl− VS

∑2l=1

1Rie

Sl

1Rie

S+ ∑2

l=11

RieSl

⎞⎟⎠

⎤⎥⎦

withdVBA

dt= 1

τm

[Rm

BAIIBA + Rm

BA

RieBA

VS2−

(1 + Rm

BA

RieBA

)VBA

].

For details regarding how the voltage difference VS2relates to the mem-

brane potential in the collector VS, we refer readers to equation D:b2 inappendix 2, section D. The equivalent resistances Rm

BA and RieBA are given by

the following general equation, with k = {m, ie}:

RkBA = Rk

BAi +Rk

BAi +Rk

BAi + RkBAt

32

4

Here, RkBAt is the kth resistance of the terminal segments, and Rk

BAi is thekth resistance of intermediate segments, both in the basal dendrites. Similarto what was observed in the oblique dendrites, these resistances can beestimated using the geometrical parameters reported by Romand et al.(2011) in the formulas developed in appendix 1. Note that for any segmenttype, l = {i, t} Rie

BAl = RiBAl + Re

BAl—the resistance RieBAl—is calculated as the

sum of the intracellular and extracellular longitudinal resistances for thatsegment.

Appendix 4. Model of the LFP

The equivalent equations for every electrotonic device have already beenobtained in appendix 1, section B; therefore, henceforth, we will use only“B: Model of the Propagator” to exemplify how to create a model of LFPs. Asimilar analysis can be applied to obtain the LFP generated by other types ofelectrotonic devices. LFPs are defined for the situation of a zero extracellularstimulation; hence, the injection current at the ECS in the equations of thepropagator were ignored.

In Figure 4, Rei represents the ith resistance of the ECS, which is located

between the ith and the i + 1th compartment, and Ui denotes the voltagedifference at the resistance Re

i . The LFP is calculated using the equation


Ui = Rei Ii. Therefore, we just need to deduct the equations of extracellular

currents, Ii. In the case of the propagator, Ii along the compartment weredefined by equations B:aN+1 to B:aN+9. Equations B:b4 to B:b8 definedthe membrane currents in the compartments; thus, we can obtain Ii at thecompartment using the following equations:

IUiT2

= IUeT2

= IUiT1

− IT1= −

VT2− VT1

RLeiT1

+ RUeiT2

(E:b1)

. . . . .

IUeTk

= IUiTk

= IUiT1

−N−1∑p=1

ITp= −

VTk+1− VTk

RLeiTk

+ RUeiTk+1

. (E:b2)

Therefore, the LFP can be defined from the extracellular potentials:UUeTk

=IUeTk

(RLeTk

+ RUeTk+1

).

Acknowledgments

We thank Juan Carlos Jimenez from the Institute of Cybernetics and Math-ematics Applied to Physics for helping us to create suitable Matlab code forintegrating large stochastic differential equations. We extend our gratitudeto Takakuni Goto for important contributions during the preparation of thiswork. We also acknowledge the help of Michael Hines and Ted Carnevalefrom Yale University in clarifying some aspects of the NEURON softwareand Alon Korngreen and Greg Stuart for sharing codes. This work was sup-ported by the JSPS Japan–Canada Joint Health Research Program (187391)“The Neuroarchitectonic Determinants of EEG Recordings;” a grant-in-aidfor Scientific Research 23300149; and a JSPS Grant-in-Aid for Young Scien-tists (B) 23700492. K.W. was supported by a JT Asia Scholarship (JT1209).

References

Angelo, K., London, M., Christensen, S. R., & Hausser, M. (2007). Local and globaleffects of I(h) distribution in dendrites of mammalian neurons. J. Neurosci., 27(32),8643–8653.

Bar-Yehuda, D., Ben-Porat, H., & Korngreen, A. (2008). Dendritic excitability dur-ing increased synaptic activity in rat neocortical L5 pyramidal neurons. Eur. J.Neurosci., 28(11), 2183–2194.

Bedard, C., & Destexhe, A. (2009). Macroscopic models of local field potentials andthe apparent 1/f noise in brain activity. Biophys. J., 96(7), 2589–2603.

Bedard, C., Rodrigues, S., Roy, N., Contreras, D., & Destexhe, A. (2010). Evidencefor frequency-dependent extracellular impedance from the transfer function be-tween extracellular and intracellular potentials: Intracellular-LFP transfer func-tion. J. Comput. Neurosci., 29(3), 389–403.


Bereshpolova, Y., Amitai, Y., Gusev, A. G., Stoelzel, C. R., & Swadlow, H. A. (2007).Dendritic backpropagation and the state of the awake neocortex. J. Neurosci.,27(35), 9392–9399.

Braitenberg, V., & Schutz, A. (1998). Cortex: Statistics and geometry of neuronal connec-tivity (2nd ed.). Berlin: Springer.

Brinkmann, B. H., Bower, M., Stengel, K. A., Worrell, G. A., & Stead, M. (2009). Large-scale electrophysiology: Acquisition, compression, encryption, and storage of bigdata. J. Neurosci. Methods, 180(1), 185–192.

Buzsaki, G. (2006). Rhythms of the brain. New York: Oxford University Press.Bush, P. C., & Sejnowski, T. J. (1993). Reduced compartmental models of neocortical

pyramidal cells. J. Neurosci. Meth., 46, 159–166.Chang, Y. M., & Luebke, J. I. (2007). Electrophysiological diversity of layer 5 pyra-

midal cells in the prefrontal cortex of the rhesus monkey: In vitro slice studies. J.Neurophysiol., 98(5), 2622–2632.

da Costa, N. M., & Martin, K. A. (2011). How thalamus connects to spiny stellatecells in the cat’s visual cortex. J. Neurosci., 31(8), 2925–2937.

Destexhe, A. (2001). Simplified models of neocortical pyramidal cells preserv-ing somatodendritic voltage attenuation. Neurocomputing (IJON), 38–40, 167–173.

Einevoll, G. T. (2010). Modeling of extracellular potentials recorded with multicon-tact microelectrodes. In Proceedings of the 7th Int. Meeting on Substrate-IntegratedMicroelectrodes. Stuttgart: BIOPRO Baden-Wurttemberg GmbH.

Gabriel, C., Gabriel, S., & Corthout, E. (1996). The dielectric properties of biologicaltissues: I. Literature survey. Phys. Med. Biol., 41(11), 2231–2249.

Gabriel, S., Lau, R. W., & Gabriel, C. (1996a). The dielectric properties of biologicaltissues: II. Measurements in the frequency range 10 Hz to 20 GHz. Phys. Med.Biol., 141(11), 2251–2269.

Gabriel, S., Lau, R. W., & Gabriel, C. (1996b). The dielectric properties of biologicaltissues: III. Parametric models for the dielectric spectrum of tissues. Phys. Med.Biol., 41(11), 2271–2293.

Gabriel, C., Peyman, A., & Grant, E. H. (2009). Electrical conductivity of tissue atfrequencies below 1 MHz. Phys. Med. Biol., 54(16), 4863–4878.

Gloveli, T., Dugladze, T., Rotstein, H. G., Traub, R. D., Monyer, H., Heinemann, U.,et al. (2005). Orthogonal arrangement of rhythm-generating microcircuits in thehippocampus. Proc. Natl. Acad. Sci. USA, 102(37), 13295–13300.

Gnatkovsky, V., Librizzi, L., Trombin, F., & de Curtis, M. (2008). Fast activity atseizure onset is mediated by inhibitory circuits in the entorhinal cortex in vitro.Ann. Neurol., 64(6), 674–686.

Gold, C., Henze, D. A., & Koch, C. (2007). Using extracellular action potentialrecordings to constrain compartmental models. J. Comput. Neurosci., 23(1), 39–58.

Gold, C., Henze, D. A., Koch, C., & Buzsaki, G. (2006). On the origin of the ex-tracellular action potential waveform: A modeling study. J. Neurophysiol., 95(5),3113–3128.

Goto, T., Hatanaka, R., Ogawa, T., Sumiyoshi, A., Riera, J., & Kawashima, R. (2010).An evaluation of the conductivity profile in the somatosensory barrel cortex ofWistar rats. J. Neurophysiol., 104(6), 3388–3412.


Gulledge, A. T., & Stuart, G. J. (2003). Action potential initiation and propagationin layer 5 pyramidal neurons of the rat prefrontal cortex: Absence of dopaminemodulation. J. Neurosci., 23(36), 11363–11372.

Harris, K. D., Henze, D. A., Csicsvari, J., Hirase, H., & Buzsaki, G. (2000). Accu-racy of tetrode spike separation as determined by simultaneous intracellular andextracellular measurements. J. Neurophysiol., 84(1), 401–414.

Henze, D. A., Borhegyi, Z., Csicsvari, J., Mamiya, A., Harris, K. D., & Buzsaki, G.(2000). Intracellular features predicted by extracellular recordings in the hip-pocampus in vivo. J. Neurophysiol., 84(1), 390–400.

Hodgkin, A., & Huxley, A. (1952). A quantitative description of membrane currentand its application to conduction and excitation in nerve. J. Physiol., 117, 500–544.

Johnston, D., & Wu, S.M-S. (1994). Foundations of cellular neurophysiology. Cambridge,MA: MIT Press.

Joshi, S., & Hawken, M. J. (2006). Loose-patch–juxtacellular recording in vivo: Amethod for functional characterization and labeling of neurons in macaque V1.J. Neurosci. Meth., 156(1–2), 37–49.

Keren, N., Peled, N., & Korngreen, A. (2005). Constraining compartmental modelsusing multiple voltage recordings and genetic algorithms. J. Neurophysiol., 94(6),3730–3742.

Kole, M. H. P., Hallermann, S., & Stuart, G. J. (2006). Single 1h channels in pyramidalneuron dendrites: Properties, distribution, and impact on action potential output.Journal of Neuroscience, 26, 1677–1687.

Korngreen, A., & Sakmann, B. (2000). Voltage-gated K+ channels in layer 5 neocor-tical pyramidal neurones from young rats: Subtypes and gradients. J. Physiol.,525(Pt. 3), 621–639.

Larkman, A. U. (1991a). Dendritic morphology of pyramidal neurones of the visualcortex of the rat: I. Branching patterns. J. Comp. Neurol., 306(2), 307–319.

Larkman, A. U. (1991b). Dendritic morphology of pyramidal neurones of the visualcortex of the rat: II. Branching patterns. J. Comp. Neurol., 306(2), 320–331.

Larkman, A. U. (1991c). Dendritic morphology of pyramidal neurones of the visualcortex of the rat: III. Branching patterns. J. Comp. Neurol., 306(2), 332–343.

Larkum, M. E., Nevian, T., Sandler, M., Polsky, A., & Schiller, J. (2009). Synaptic inte-gration in tuft dendrites of layer 5 pyramidal neurons: A new unifying principle.Science, 325(5941), 756–760.

Lehmenkuhler, A., Sykova, E., Svoboda, J., Zilles, K., & Nicholson, C. (1993). Ex-tracellular space parameters in the rat neocortex and subcortical white matterduring postnatal development determined by diffusion analysis. Neuroscience,55(2), 339–351.

Linden, H., Pettersen, K. H., & Einevoll, G. T. (2010). Intrinsic dendritic filteringgives low-pass power spectra of local field potentials. J. Comput. Neurosci., 29(3),423–444.

Lytton, W. W., & Sejnowski, T. J. (1991). Simulations of cortical pyramidal neuronssynchronized by inhibitory interneurons. J. Neurophysiol., 66(3), 1059–1079.

Mainen, Z. F., Joerges, J., Huguenard, J. R., & Sejnowski, T. J. (1995). A model of spikeinitiation in neocortical pyramidal neurons. Neuron, 15(6), 1427–1439.

Milstein, J. N., & Koch, C. (2008). Dynamic moment analysis of the extracellularelectric field of a biologically realistic spiking neuron. Neural Comput., 20(8),2070–2084.


Murakami, S., & Okada, Y. (2006). Contributions of principal neocortical neurons tomagnetoencephalography and electroencephalography signals. J. Physiol., 575(Pt.3), 925–936.

Neher, E. (1971). Two fast transient current components during voltage clamp onsnail neurons. J. Gen. Physiol., 158(1), 36–53.

Pearce, J. M. (2001). Emil Heinrich Du Bois-Reymond (1818–96). J. Neurol. Neurosurg.Psychiatry, 71(5), 620.

Pettersen, K. H., Devor, A., Ulbert, I., Dale, A. M., & Einevoll, G. T. (2006). Current-source density estimation based on inversion of electrostatic forward solution:Effects of finite extent of neuronal activity and conductivity discontinuities. J.Neurosci. Methods, 154(1–2), 116–133.

Pinault, D. (2008). The juxtacellular recording-labeling technique. Neuromethods, 54,41–75.

Plonsey, R., & Heppner, D. B. (1967). Considerations of quasi-stationarity in electro-physiological systems. Bull. Math. Biophys., 29(4), 657–664.

Rall, W. (1957). Membrane time constant of motoneurons. Science, 126, 454.Rall, W. (1959). Branching dendritic trees and motoneuron membrane resistivity.

Exp. Neurol., 1, 491–527.Rall, W. (1960). Membrane potential transients and membrane time constant of mo-

toneurons. Exp. Neurol., 2, 503–532.Rall, W. (1962). Electrophysiology of a dendritic neuron model. Bioph. J., 2, 145–

167.Rall, W. (1964). Theoretical significance of dendritic trees for neuronal input-output

relations. In R.F. Reiss (Ed.), Neural theory and modeling. Stanford, CA: StanfordUniversity Press.

Rhodes, P. (2006). The properties and implications of NMDA spikes in neocorticalpyramidal cells. J. Neurosci., 26(25), 6704–6715.

Riera, J., Jimenez, J. C., Wan, X., Kawashima, R., & Ozaki, T. (2007). Nonlinear localelectro-vascular coupling: part II: From data to neural masses. Hum. Brain Mapp.,28, 335–354.

Riera, J., Ogawa, T., Goto, T., Sumiyoshi, A., Nonaka, H., Evans, A., et al. (2012).Pitfalls in the dipolar model for the neocortical EEG sources. J. Neurophysi., 108,956–975.

Riera, J. J., Wan, X., Jimenez, J. C., & Kawashima, R. (2006). Nonlinear local elec-trovascular coupling. I: A theoretical model. Hum. Brain Mapp., 27(11), 896–914.

Rodriguez, A., Ehlenberger, D., Kelliher, K., Einstein, M., Henderson, S. C., Mori-son, J. H., et al. (2003). Automated reconstruction of three-dimensional neuronalmorphology from laser scanning microscopy images. Methods, 30, 94–105.

Romand, S., Wang, Y., Toledo-Rodriguez, M., & Markram, H. (2011). Morphologicaldevelopment of thick-tufted layer V pyramidal cells in the rat somatosensorycortex. Frontiers in Neuroanatomy, 5(5), 1–25.

Sakmann, B., & Neher, E. (1984). Patch clamp techniques for studying ionic channelsin excitable membranes. Annu. Rev. Physiol., 46, 455–472.

Schaefer, A. T., Helmstaedter, M., Schmitt, A. C., Bar-Yehuda, D., Almog, M., Ben-Porat, H., et al. (2007). Dendritic voltage-gated K+ conductance gradient in pyra-midal neurons of neocortical layer 5B from rats. J. Physiol., 579(3), 737–752.

Somogyvari, Z., Zalanyi, L., Ulbert, I., & Erdi, P. (2005). Model-based source local-ization of extracellular action potentials. J. Neurosci. Meth., 147(2), 126–137.


Stark, E., & Abeles, M. (2007). Predicting movement from multiunit activity. J. Neu-rosci., 27(31), 8387–8394.

Stratford, K., Mason, A., Larkman, A., Major, G., & Jack, J. (1989). The modelling ofpyramidal neurones in the visual cortex. In R. Durbin, C. Miall, & G. Mitchison(Eds.), The computing neuron (pp. 296–321). Reading, MA: Addison-Wesley.

Stuart, G. J., & Sakmann, B. (1994). Active propagation of somatic action potentialsinto neocortical pyramidal cell dendrites. Nature, 367(6458), 69–72.

Stuart, G., & Sakmann, B. (1995). Amplification of EPSPs by axosomatic sodiumchannels in neocortical pyramidal neurons. Neuron, 15(5), 1065–1076.

Swadlow, H. A., Gusev, A. G., & Bezdudnaya, T. (2002). Activation of a corticalcolumn by a thalamocortical impulse. J. Neurosci., 22(17), 7766–7773.

Sykova, E., & Nicholson, C. (2008). Diffusion in brain extracellular space. Physiol.Rev., 88(4), 1277–1340.

Traub, R. D., Contreras, D., Cunningham, M. O., Murray, H., LeBeau, F.E.N., Roopun,A., et al. (2005). Single-column thalamocortical network model exhibiting gammaoscillations, sleep spindles, and epileptogenic bursts. J. Neurophysiol., 93, 2194–2232.

Tuckwell, H. C. (1988). Introduction to theoretical neurobiology. Cambridge: CambridgeUniversity Press.

Wilent, W. B., Oh, M. Y., Buetefisch, C., Bailes, J. E., Cantella, D., Angle, C., et al.(2011). Mapping of microstimulation evoked responses and unit activity patternsin the lateral hypothalamic area recorded in awake humans. J. Neurosurg., 115(2),295–300.

Wilson, D. A. (2010). Single-unit activity in piriform cortex during slow-wave stateis shaped by recent odor experience. J. Neurosci., 30(5), 1760–1765.

Received June 28, 2011; accepted November 6, 2012.

The Role of Extracellular Conductivity Proﬁles in ...

Documents

Transcript of The Role of Extracellular Conductivity Proﬁles in ...