Computing extracellular electric potentials from neuronal simulations
Torbjørn V. Ness, Geir Halnes, Solveig Næss, Klas H. Pettersen, Gaute T. Einevoll
CComputing extracellular electric potentials from neuronal simulations
Torbjørn V. Ness a, ∗ , Geir Halnes a, ∗ , Solveig Næss b , Klas H. Pettersen c , Gaute T. Einevoll a,d, ∗∗ a Faculty of Science and Technology, Norwegian University of Life Sciences, Ås, Norway b Department of Informatics, University of Oslo, Oslo, Norway c Norwegian Artificial Intelligence Research Consortium, Oslo, Norway d Department of Physics, University of Oslo, Oslo, Norway
Abstract
Measurements of electric potentials from neural activity have played a key role in neurosciencefor almost a century, and simulations of neural activity is an important tool for understanding suchmeasurements. Volume conductor (VC) theory is used to compute extracellular electric potentialssuch as extracellular spikes, MUA, LFP, ECoG and EEG surrounding neurons, and also inversely,to reconstruct neuronal current source distributions from recorded potentials through current sourcedensity methods. In this book chapter, we show how VC theory can be derived from a detailedelectrodiffusive theory for ion concentration dynamics in the extracellular medium, and show whatassumptions that must be introduced to get the VC theory on the simplified form that is commonlyused by neuroscientists. Furthermore, we provide examples of how the theory is applied to computespikes, LFP signals and EEG signals generated by neurons and neuronal populations.
Keywords: extracellular potentials, LFP, EEG, ECoG, electrodiffusion, neuronal simulation, MUA
Contents1 Introduction 22 From electrodiffusion to volume conductor theory 3 ∗ These authors have contributed equally to this work ∗∗ correspondence: [email protected] a r X i v : . [ q - b i o . N C ] J un .2 MUA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Arguably, most of what we have learned about the mechanisms by which neurons and net-works operate in living brains comes from recordings of extracellular potentials. In such recordings,electric potentials are measured by electrodes that are either placed between cells in brain tissue(spikes, LFPs), at the cortical surface (ECoG, electrocorticography), or at the scalp (EEG, electroen-cephalograpy) (Figure 1). Spikes are reliable signatures of neuronal action potentials, and spikemeasurements have been instrumental in mapping out, for example, receptive fields accounting forhow sensory stimuli are represented in the brain. The analysis of the LFP signal, the low-frequencypart of electric potentials recorded inside gray matter, as well as the ECoG, and EEG signals is lessstraightforward. Interpretation of these signals in terms of the underlying neural activity has beendifficult, and most analyses of the data have been purely statistical [Nunez and Srinivasan, 2006;Buzsáki et al., 2012; Einevoll et al., 2013a; Ilmoniemi and Sarvas, 2019].The tradition of physics is different. Here candidate hypotheses are typically formulated as spe-cific mathematical models, and predictions computed from the models are compared with experi-ments. In neuroscience this approach has been used to model activity in individual neurons using,for example, biophysically detailed neuron models based on the cable equation formalism (see, e.g.,Koch [1999]; Sterratt et al. [2011]). These models have largely been developed and tested by com-parison with membrane potentials recorded by intracellular electrodes in in vitro settings (but seeGold et al. [2007]). To pursue this mechanistic approach to network models in layered structuressuch as cortex or hippocampus, one would like to compare model predictions with all available ex-perimental data, that is, not only spike times recorded for a small subset of the neurons, but alsopopulation measures such as LFP, ECoG and EEG signals [Einevoll et al., 2019]. This chapteraddresses how to model such electric population signals from neuron and network models.In addition to allowing for validation on large-scale network models mimicking specific biologi-cal networks, e.g., Reimann et al. [2013]; Markram et al. [2015]; Billeh et al. [2020], we believe akey application is to generate model-based benchmarking data for validation of data analysis meth-ods [Denker et al., 2012]. One example is the use of such benchmarking data to develop and testspike-sorting methods Hagen et al. [2016]; Buccino and Einevoll [2019] or test methods for localiza-tion and classification of cell types [Delgado Ruz and Schultz, 2014; Buccino et al., 2018]. Anotherexample is testing of methods for analysis of LFP signals, such as CSD analysis [Pettersen et al.,2008; Łe¸ ski et al., 2011; Ness et al., 2015] or ICA analysis [Gł ˛abska et al., 2014], or joint analysis ofspike and LFP signals such as laminar population analysis (LPA) [Gł ˛abska et al., 2016].The standard way to compute extracellular potentials from neural activity is a two-step pro-cess [Holt and Koch, 1999; Lindén et al., 2014; Hagen et al., 2018]: Compute the net transmembrane current in all neuronal segments in (networks of) biophysically-detailed neuron models, and 2 . use volume-conductor (VC) theory to compute extracellular potentials from the these com-puted transmembrane currents.In this book chapter we describe the origin of VC theory, that is, how it can be derived from amore detailed electrodiffusive theory describing dynamics of ions in the extracellular media. Wefurther provide examples where our tool LFPy ( LFPy.github.io ) [Lindén et al., 2014; Hagen et al.,2018] is used to compute spikes, LFP signals and EEG signals generated by neurons and neuronalpopulations.
Figure 1:
LFP, ECoG and EEG.
The same basic building blocks, that is, currents caused by large numbers of synapticinput are contributing to several different measurable signals.
2. From electrodiffusion to volume conductor theory
Extracellular potentials are generated by electric currents in the extracellular space. The currentsare in turn mediated by movement of ions, and can in principle include several components: a drift component (ions migrating in electric fields), a diffusion component (ions diffusing due to concentration gradients), an advective component (extracellular fluid flow drags ions along), and a displacement component (ions pile up and changes the local charge density).Since the extracellular bulk fluid has very fast relaxation times and is very close to electroneutral,the latter two current components (3-4) are extremely small and are typically neglected [Grodzinsky,2011; Gratiy et al., 2017]. The diffusive component (2) is acknowledged to play an important role forvoltage dynamics on a tiny spatial scale, such as in synaptic clefts or in the close vicinity of neuronalmembranes, where ion concentrations can change dramatically within very short times [Holcman3nd Yuste, 2015; Savtchenko et al., 2017; Pods, 2017]. At macroscopic tissue level, it is commonlyassumed that the diffusive current is much smaller than the drift current, so that in most studies, onlythe drift component (1) is considered. The extracellular medium can then be treated as a volumeconductor (VC), which greatly simplifies the calculation of extracellular potentials [Holt and Koch,1999; Lindén et al., 2014].However, if large ion-concentration gradients are present, diffusive currents could in principleaffect measurable extracellular potentials [Halnes et al., 2016, 2017; Solbrå et al., 2018]. Thus inscenarios involving dramatic shifts in extracellular concentrations, such as spreading depression andrelated pathologies, diffusive effects are likely to be of key importance for shaping the extracellularpotential [Almeida et al., 2004; O’Connell and Mori, 2016]. For such cases VC theory is insufficient,and computationally much more expensive electrodiffusive modeling must be used.In this section the starting point is the general assumption of ion movement under the combinedinfluence of electric fields and concentration gradients. Building on this, we first describe compu-tational schemes for modelling electrodiffusive processes, and next show how the electrodiffusivetheory reduces to the fundamental equations for VC theory when we assume negligible effects fromdiffusion. The movement of ions in the brain are described in terms of fluxes. In electrodiffusive processes,the flux density of an ion species k is given by [Koch, 1999]: j k = − D k ∇ c k − D k z k c k ψ ∇ φ, (1)where the first term on the right is Fick’s law for the diffusive flux density j diff k , and the second termis the drift flux density j drift k , which expands Fick’s law in the case where the diffusing particles alsomove due to electrostatic forces with a mobility D k /ψ (cf. the Einstein-relation, Mori et al. [2008]).Here D k is the diffusion coefficient of ion species k , φ is the electric potential, z k is the valency ofion species k , and ψ = RT /F is defined by the gas constant ( R ), Faraday’s constant ( F ) and thetemperature ( T ). The ion concentration dynamics of a given species is then given by the Nernst-Planck continuity equation, ∂c k ∂t = −∇ · j k + f k = ∇ · (cid:20) D k ∇ c k + D k z k c k ψ ∇ φ (cid:21) + f k , (2)where f k represents any source term in the system, such as e.g., an ionic transmembrane currentsource [Solbrå et al., 2018].In order to solve a set (i.e., one for each ion species present) of equations like eq. 2, one needsan expression for the electric potential φ . There are two main approaches to this. The physicallymost detailed approach is to use the Poisson-Nernst-Planck (PNP) formalism [Léonetti and Dubois-Violette, 1998; Léonetti et al., 2004; Lu et al., 2007; Lopreore et al., 2008; Nanninga, 2008; Podset al., 2013; Gardner et al., 2015; Cartailler et al., 2018]. Within this formalism, φ is determined fromPoisson’s equation from electrostatics, ∇ φ = − ρ/(cid:15), (3)4here (cid:15) is the permittivity of the system, and ρ is the charge density associated with the ionicconcentrations, as given by ρ = F (cid:88) k z k c k . (4)An alternative, more computationally efficient approach is to replace the Poisson equation with thesimplifying approximation that the bulk solution is electroneutral [Mori et al., 2008; Mori, 2009; Moriand Peskin, 2009; Mori et al., 2011; Niederer, 2013; Halnes et al., 2013, 2015; Pods, 2017; O’Connelland Mori, 2016; Solbrå et al., 2018; Tuttle et al., 2019; Ellingsrud et al., 2020; Sætra et al., 2020],which is a good approximation on spatiotemporal scales larger than micrometers and microseconds[Grodzinsky, 2011; Pods, 2017; Solbrå et al., 2018].Both the PNP formalism and the electroneutral formalism allow us to compute the dynamicsof ion concentrations and the electric potential in the extracellular space of neural tissue contain-ing an arbitrary set of neuronal and glial current sources. For example, in recent work, a versionof the electroneutral formalism called the Kirchhoff-Nernst-Planck (KNP) formalism was developedinto a framework for computing the extracellular dynamics (of c k and φ ) in a 3D space surroundingmorphologically complex neurons simulated with the NEURON simulation tool [Solbrå et al., 2018].However, both the PNP and electroneutral formalisms such as KNP keep track of the spatial dis-tribution of ion concentrations, and as such they require a suitable meshing of the 3D space, andnumerical solutions based on finite difference- or finite element methods. In both cases, simulationscan become computationally demanding, and for systems at a tissue level the required compu-tational demand may become unfeasible. For that reason, there is much to gain from deriving asimpler framework where effects of ion concentration dynamics are neglected, and for many sce-narios this may be a good approximation. Below, we will derive this simpler framework, i.e., thestandard volume conductor (VC) theory, using the Nernst-Planck fluxes (eq. 1) as a starting point. If we multiply eq. 1 by F · z k and sum over all ion species, we get an expression for the netelectric current density due to all particle fluxes, i = − (cid:88) k F z k D k ∇ c k − σ ∇ φ, (5)where the first term is the diffusive current density i diff and the second term is the drift current density i drift . We have here identified the conductivity σ of the medium as [Koch, 1999]: σ = F (cid:88) k ˜ D k z k ψ c k . (6)Current conservation in the extracellular space implies that: ∇ · i = − (cid:88) k F z k D k ∇ c k − ∇ · ( σ ∇ φ ) = − C, (7)where C denotes the current source density (CSD), reflecting e.g., local neuronal or glial transmem-brane currents. We note that this is essentially equivalent to eq. 2 at the level of single ion species,5ith the exception that eq. 2 contains a term ∂c k /∂t for accumulation of ion species k , while eq. 7does not contain a corresponding term ( ∂ρ/∂t ) for charge accumulation. Hence, in eq. 7 it is im-plicitly assumed that the extracellular bulk solution is electroneutral [Solbrå et al., 2018]. We notethat in general, the CSD term includes both ionic transmembrane currents and transmembranecapacitive currents, and that the latter means that the local charge accumulation building up thetransmembrane potential still occurs in the membrane Debye-layer.Note that if we assume all concentrations to be constant in space, the diffusive term vanishes,and eq. 7 reduces to ∇ · ( σ ∇ φ ) = − C. (8)This the standard expression used in CSD theory [Mitzdorf, 1985; Nicholson and Freeman, 1975;Pettersen et al., 2006], where spatially distributed recordings of φ are used to make theoretical pre-dictions of underlying current sources. When using eq. 8, it is implicitly assumed that the Laplacianof φ exclusively reflects transmembrane current sources, and that it is not contributed to by diffusiveprocesses.Note that there are two commonly used conventions for defining the variables in eqns. 1-8. Thevariables can be defined either relative to a tissue reference volume or relative to an extracellularreference volume. The former convention is the common convention used in volume conductortheory. For this convention, concentrations denote the number of extracellular ions per unit tissuevolume, sources denote the number of ions or the net charge per unit tissue volume per second,and flux or current densities are defined per unit tissue cross-section area. Finally, σ interpretsas the tissue-averaged extracellular conductivity, i.e., it is not the conductivity of the extracellularsolution as such, but accounts for the fact that extracellular currents at the coarse-grained scale (i)have tortuous trajectories around neural and glial obstacles, and (ii) are mostly confined to moveonly through the extracellular fraction (typically about 0.2) of the total tissue volume [Nicholson andSyková, 1998; Nunez and Srinivasan, 2006].Note that there are two commonly used conventions for defining the variables in eqns. 1-8. Thevariables can be defined either relative to a tissue reference volume or relative to an extracellularreference volume. The former convention is the common convention used in volume conductortheory. For this convention, concentrations denote the number of extracellular ions per unit tissuevolume, sources denote the number of ions or the net charge per unit tissue volume per second,and flux or current densities are defined per unit tissue cross-section area. Finally, σ interpretsas the tissue-averaged extracellular conductivity, i.e., it is not the conductivity of the extracellularsolution as such, but accounts for the fact that extracellular currents at the coarse-grained scale (i)are mostly confined to move only through the extracellular fraction (typically about 0.2) of the totaltissue volume, and (ii) must take detours around neural and glial obstacles [Nicholson and Syková,1998; Nunez and Srinivasan, 2006].As eq. 7 indicates, also diffusive processes can in principle contribute to the Laplacian of φ , andif present, they could give rise to a non-zero Laplacian of φ even in the absence of neuronal sources( C = 0 ). Previous computational studies have predicted that effects of diffusion on extracellularpotentials are not necessarily small, but tend to be very slow, meaning that they will only affectthe very low-frequency components of φ [Halnes et al., 2016, 2017]. This is due to the diffusivecurrent being a direct function of ion concentrations c k , which on a large spatial scale typically varyon a much slower time scale (seconds to minutes) than the fluctuations in φ that we commonly6re interested in (milliseconds to seconds). Furthermore, electrodes used to record φ typicallyhave a lower cutoff frequency between 0.1 and 1 Hz [Einevoll et al., 2013a], which means thatmost of the tentative diffusive contribution will be filtered out from experimental recordings. It maytherefore be a good approximation to neglect the diffusive term, except in the case of pathologicallydramatic concentration variations. For the rest of this chapter, we shall do so, and assume thatelectrodynamics in neural tissue can be determined by eq. 8. In simulations of morphologically complex neurons, one typically computes a set of transmem-brane current sources for each neuronal segment [Koch, 1999]. By assuming that the tissue mediumcan be approximated as a volume conductor [Holt and Koch, 1999; Lindén et al., 2014], one can thenuse the standard CSD equation (eq. 8) to perform a forward modeling of the extracellular potentialat each point in space surrounding the neuron(s). Since extracellular potentials are generally muchsmaller than the membrane potential of ∼ -70 mV, it is common to assume that the neurodynamicsis not affected by extracellular potentials, and to simulate the neurodynamics as a first independentstep, before computing the extracellular potentials in the next step.If we consider the simple case of a single point-current source I at the origin in an isotropicmedium, the current density i = − σ ∇ φ through a spherical shell with area πr must, due to thespherical symmetry, equal I / πr ˆr . Integration with respect to r gives us: φ = I πσr , (9)where r is the distance from the source.If we have several point-current sources, I , I , I , ... , in locations r , r , r ... , their contributionsadd up due to the linearity assumption (see sec. 2.3.2), and the potential in a point r is given by: φ ( r ) = I πσ | r − r | + I πσ | r − r | + I πσ | r − r | + ... = (cid:88) k I k πσ | r − r k | . (10)Eq. 10 is often referred to as the point-source approximation [Holt and Koch, 1999; Lindén et al.,2014], since the membrane current from a neuronal segment is assumed to enter the extracellularmedium in a single point. An often used further development is obtained by integrating eq. 10 alongthe segment axis, corresponding to the transmembrane current being evenly distributed along thesegment axis, giving the line-source approximation [Holt and Koch, 1999; Lindén et al., 2014]. When estimating the extracellular potential far away from a volume containing a combination ofcurrent sinks and sources, it can often be useful to express eq. (10) in terms of a multipole expansion.That is, φ can be precisely described by [Nunez and Srinivasan, 2006], φ ( R ) = C monopole R + C dipole R + C quadrupole R + C octupole R + ..., (11)when the distance R from the center of the volume to the measurement point is larger than thedistance from volume center to the most peripheral source [Jackson, 1998].7n neural tissue, there will be no current monopole contribution to the extracellular potential, C monopole = 0 . This follows from the requirement inherent in the cable equation that the sum overall transmembrane currents, including the capacitive currents, across the neuronal surface has tobe zero at all points in time [Pettersen et al., 2012]. Further, the quadrupole, octupole and higher-order contributions decay rapidly with distance R . Consequently, the multipole expansion can beapproximated by the dipole contribution for large distances, a simplification known as the current-dipole approximation [Nunez and Srinivasan, 2006]: φ ( R ) ≈ C dipole R = 14 πσ | p | cos θR . (12)Here, p is the current dipole moment and θ is the angle between the current dipole moment and thedistance vector R . The current dipole moment can be found by summing up all the position-weightedtransmembrane currents from a neuron [Pettersen et al., 2008, 2014; Nunez and Srinivasan, 2006]: p = N (cid:88) k =1 I k r k . (13)In the case of a two-compartment neuron model (see Section 3) with a current sink − I at location r and a current source I at location r , the current dipole moment can be formulated as p = − I r + I r = I ( r − r ) = I d , where d is the distance vector between the current sink and thecurrent source, giving the dipole length d and direction of the current dipole. The current-dipoleapproximation is applicable in the far-field limit, that is when R is much larger than the dipole length.For an investigation of the applicability of this approximation for the LFP generated by a singleneuron, see Lindén et al. [2010]. The point-source approximation, eq. 10 (or the line-source version of it), and the current-dipoleapproximation, eq. (12), represent volume conductor theory in its simplest form, and are based on aset of assumptions, some of which may be relaxed for problems where it is relevant: Quasi-static approximation of Maxwell’s equations:
Terms with time derivatives of theelectric and magnetic fields are neglected. This approximation appears to be well-justifiedfor the relatively low frequencies relevant for brain signals, below about 10 kHz [Nunez andSrinivasan, 2006]. Linear extracellular medium:
Linear relationship ( i = − σ ∇ φ ) between the current density i and the electric field, ∇ φ . This is essentially Ohm’s law for volume conductors, and therelation is constitutive, meaning that it is observed in nature rather than derived from anyphysical principle [Nunez and Srinivasan, 2006; Pettersen et al., 2012]. Frequency-independent conductivity:
Capacitive effects in neural tissue are assumed to benegligible compared to resistive effects in volume conduction. This approximation seems to bejustified for the relevant frequencies in extracellular recordings [Logothetis et al., 2007; Miceliet al., 2017; Ranta et al., 2017], see Fig. 2. Note that it is possible to expand the formalism toinclude a frequency-dependent conductivity [Tracey and Williams, 2011; Miceli et al., 2017].8 . Isotropic conductivity:
The electric conductivity, σ , is assumed to be the same in all spatialdirections. Cortical measurements have indeed found the conductivities to be comparableacross different lateral directions in cortical grey matter [Logothetis et al., 2007]. However,the conductivity in the depth direction, i.e., parallel to the long apical dendrites, was foundto be up to 50% larger than in the lateral direction in rat barrel cortex [Goto et al., 2010].Anisotropic electric conductivities have also been found in other brain regions, for example infrog cerebellum [Nicholson and Freeman, 1975] and in guinea-pig hippocampus [Holsheimer,1987]. The approximation that σ is homogeneous is still often acceptable, as it normally givesfairly good estimates of the extracellular potential, at least in cortical tissue [Ness et al., 2015].However, it is relatively straightforward to expand the formalism to account for anisotropicconductivities [Ness et al., 2015]. Homogeneous conductivity:
The extracellular medium was assumed to have the same con-ductivity everywhere. Although neural tissue is highly non-homogeneous on the micrometerscale [Nicholson and Syková, 1998], microscale inhomogeneities may average out on a largerspatial scale, and a homogeneous conductivity seems to be a reasonable approximation withincortex [Logothetis et al., 2007]. In hippocampus, however, the conductivity has been foundto be layer-specific [López-Aguado et al., 2001]. In situations where the assumption of ahomogeneous conductivity is not applicable, eq. 8 can always be solved for arbitrarily com-plex geometries using numerical methods, like the Finite Element Method (FEM) [Logg et al.,2012]. For some example neuroscience applications, see Moffitt and McIntyre [2005]; Freyet al. [2009]; Joucla and Yvert [2012]; Haufe et al. [2015]; Ness et al. [2015]; Buccino et al.[2019]; Obien et al. [2019]. For some simple non-homogeneous cases analytical solutions canstill be obtained, for example through the Method of Images for in vitro brain slices [Ness et al.,2015], and the four-sphere head model for EEG signals (Sec. 5) [Næss et al., 2017]. No effects from ion diffusion:
To account for diffusion of ions, one would need to computethe electrodynamics of the system using one of the electrodiffusive frameworks presented inSection 2.1.Volume conductor theory is the fundament for forward modeling of extracellular potentials atdifferent spatial scales, from extracellular spikes, LFPs and MUAs, to ECoGs and EEGs. In thefollowing sections we shall review previous modeling works, and insights from simulating electricpotentials at these different scales. We use the software LFPy [Lindén et al., 2014; Hagen et al.,2018, 2019], which has volume conductor theory incorporated and can in principle be used to com-pute extracellular potentials on arbitrarily large spatial scales, surrounding arbitrarily large neuronalpopulations.
The simplest and most commonly used approach when modeling extracellular recordings is tocalculate the extracellular potential at single points following one of the approaches outlined above,and use this as a measure of recorded potentials. Implicitly, this assumes ideal point electrodes,that is, the electrodes (and electrode shank) do not affect the extracellular potential and the extra-cellular potential does not vary substantially over the surface of the electrodes. (The point-electrodeassumption was used for all simulation examples in this chapter).9 igure 2:
Literature review of reported conductivities in various species and experimental setups.
Most studiesseem to indicate a very weak frequency dependence of the extracellular conductivity, which would have a negligibleeffect on measured extracellular potentials [Miceli et al., 2017]. The very low and strongly frequency dependent valuesmeasured by Gabriel et al. [1996] represents an outlier, and although it has received substantial attention, it has to thebest of our knowledge not been reproduced by any other study. For details about the data, see [Miceli et al., 2017], andreferences therein [Ranck, 1963; Gabriel et al., 1996; Logothetis et al., 2007; Elbohouty, 2013; Wagner et al., 2014].
A numerically straightforward extension is the disc-electrode approximation where the potentialis evaluated at a number of points on the electrode surface, and the average calculated [Lindénet al., 2014]. This approach takes into account the physical extent of the electrode, but not anyeffect the electrode itself might have on the electric potential. Close to the electrode surface theelectric potential will however be affected by the presence of the high-conductivity electrode contact[McIntyre and Grill, 2001; Moulin et al., 2008]. A numerically much more comprehensive approach tomodeling electrodes is to use the Finite Element Method (FEM) to model the electrode [Moulin et al.,2008; Ness et al., 2015], or the electrode shank [Moffitt and McIntyre, 2005; Buccino et al., 2019].Using FEM for validation, Ness et al. [2015] found that the ideal point-electrode and disc-electrodeapproximations where reasonably accurate when the distance between the current sources and therecording electrode was bigger than ∼ ∼ . Single-cell contributions to extracellular potentials The transmembrane currents of a neuron during any neural activity can be used to calculateextracellular potentials, by applying the formalism described in Sec. 2.3, and in the simplest caseeq. 10. Current conservation requires that the transmembrane currents across the entire cellularmembrane at any given time sum to zero [Koch, 1999; Nunez and Srinivasan, 2006], and sincean excitatory synaptic input generates a current sink (negative current), this will necessarily lead tocurrent sources elsewhere on the cell. This implies that point neurons, that is, neurons with no spatialstructure, will have no net transmembrane currents, and hence cause no extracellular potentials(Fig. 3A). The simplest neuron models that are capable of producing extracellular potentials aretherefore two-compartment models, which will have two equal but opposite transmembrane currents,giving rise to perfectly symmetric extracellular potentials (Fig. 3B).Multi-compartment neuron models mimicking the complex spatial structure of real neurons willtypically give rise to complicated patterns of current sinks and sources, leading to complex, butmostly dipolar-like extracellular potentials (Fig. 3C) [Einevoll et al., 2013a]. Note that this frame-work for calculating extracellular potentials is valid both for subthreshold and suprathreshold neuralactivity, that is, when a cell receives synaptic input that does not trigger, or does trigger an actionpotential, respectively (Fig. 3, D versus E).
4. Intra-cortical extracellular potentials from neural populations
Extracellular potentials measured within neural tissue are often split into two separate frequencydomains, which reflect different aspects of the underlying neural activity. The low frequency part,the local field potential (LFP), is thought to mostly reflect synaptic input to populations of pyramidalcells, while the high-frequency part, the multi-unit activity (MUA), reflects the population spikingactivity (Fig. 4).
The LFP is the low-frequency part ( (cid:46)
500 Hz) of the extracellular potentials, and it is among theoldest and most used brain signals in neuroscience [Einevoll et al., 2013a]. The LFP is expectedto be dominated by synaptic inputs asymmetrically placed onto populations of geometrically alignedneurons [Nunez and Srinivasan, 2006; Lindén et al., 2011; Einevoll et al., 2013b]. In cortex andhippocampus, neurons can broadly speaking be divided into two main classes: the inhibitory in-terneurons, and the excitatory pyramidal neurons. Pyramidal neurons typically have a clear axis oforientation, that is, the apical dendrites of close-by pyramidal neurons tend to be oriented in the samedirection (Fig. 4A). This geometrical alignment is important because the LFP contributions from theindividual pyramidal cells also align and therefore sum constructively. For example, basal excita-tory synaptic input (Fig. 4B, time marked by red line) generates a current sink and correspondingnegative LFP deflection in the basal region, and simultaneously a current source and correspondingpositive LFP deflection in the apical region (Fig. 4D, time marked by red line), while apical excita-tory synaptic input leads to the reversed pattern (Fig. 4B,D, time marked by blue line). Importantly,this means that excitatory input that simultaneously targets both the apical and the basal dendritewill give opposite source/sink patterns which will lead to substantial cancellation and a weak LFP11 igure 3:
Single-cell contributions to the extracellular potential. A : Point neurons have no net currents (top),and therefore cause no extracellular potentials (bottom). B : Two-compartment neuron models have two opposite cur-rents of identical magnitude (top), and cause perfectly symmetric dipolar-like extracellular potentials (bottom). C : Multi-compartment neuron models [Hay et al., 2011] give rise to complex source-sink patterns (top) and complex (but mostlydipolar-like) extracellular potentials (bottom). D, E : A single somatic synaptic input to a complex multi-compartment cellmodel, either subthreshold (D) or suprathreshold (E; double synaptic weight of D), illustrating that the same framework canbe used to calculate both the extracellular potential from subthreshold synaptic input, and extracellular action potentials. contribution (Fig. 4B, D, time marked by orange line). The same arguments also apply to inhibitorysynaptic inputs, with the signs of the currents and LFPs reversed.Note that, for example, the LFP signature of apical excitatory synaptic input is inherently similarto that of basal inhibitory input, and indeed, separating between cases like this pose a real challengein interpreting LFP signals [Lindén et al., 2010].In contrast to pyramidal neurons, interneurons often lack any clear orientational specificity, mean-ing that the current dipoles from individual interneurons, which might by themselves be sizable[Lindén et al., 2010], do not align, leading to negligible net contributions to LFP signals [Mazzoniet al., 2015]. Note, however, that the interneurons may indirectly cause large LFP contributionsthrough their synaptic inputs onto pyramidal cells [Tele ´nczuk et al., 2017; Hagen et al., 2016].It has been demonstrated that correlations among the synaptic inputs to pyramidal cells can12 igure 4:
Extracellular potentials from different waves of synaptic input . Different brain signals from separate wavesof excitatory synaptic input to 10 000 layer 5 pyramidal cells from rat [Hay et al., 2011]. A : A subset of 100 pyramidal cells,with the LFP electrode locations indicated in the center (colored dots). B : Depth-resolved synaptic inputs arrive in threewaves, first targeting the basal dendrites (t=100 ms), then the apical dendrites (t=200 ms), and lastly uniformly across theentire depth (t=300 ms). Note that all synaptic input is pre-defined, that is, there is no network activity. C : The extracellularpotential at different depths (corresponding to dots in panel A), including both spikes and synaptic input. D : The LFP, thatis, a low-pass filtered version of the raw signal in C. E : The MUA, that is, a high-pass filtered version of the raw signal inC. F : Another version of the MUA which is a rectified and low-pass filtered version of the MUA signal in E. All filters were4th order Butterworth filters in forward-backward mode [NeuroEnsamble, 2017]. For illustrative purposes a relatively lowcut-off frequency of 50 Hz was chosen for the LFP low-pass filter. The MUA was first high-pass filtered above 300 Hz (Eand F), then rectified and low-pass filtered below 300 Hz (F). amplify the LFP signal power by orders of magnitude, with the implication that populations receivingcorrelated synaptic input can dominate the LFP also 1-2 mm outside of the population [Lindén et al.,2011; Łe¸ ski et al., 2013].Somatic action potentials lasting only a few milliseconds are generally expected to contribute littleto cortical LFP signals [Pettersen et al., 2008; Pettersen and Einevoll, 2008; Einevoll et al., 2013a;Haider et al., 2016]: Their very short duration with both positive and negative phases (Fig. 3E)will typically give large signal cancellations of the contributions from individual neurons, and theirhigh frequency content is to a large degree removed from LFPs during low-pass filtering. Note,however, that in the hippocampus the highly synchronized spikes found during sharp wave ripplesare expected to also contribute to shaping of the LFP [Schomburg et al., 2012; Luo et al., 2018].Other active conductances may contribute in shaping the LFP, for example, the slower dendriticcalcium spikes [Suzuki and Larkum, 2017] or long-lasting after-hyperpolarization currents [Reimannet al., 2013]. Further, subthreshold active conductances can also shape the LFP by molding thetransmembrane currents following synaptic input, and the hyperpolarization-activated cation channel13 h may play a key role in this, both through asymmetrically changing the membrane conductance,and by introducing apparent resonance peaks in the LFP [Ness et al., 2016, 2018]. While LFPs are thought to mainly reflect the synaptic input to large populations of pyramidalneurons, the multi-unit activity (MUA) can be used to probe the population spiking activity [Einevollet al., 2007; Pettersen et al., 2008] (Fig. 4 E,F). In other words, the MUA holds complimentaryinformation to the LFP. In particular, this can be useful for some cell-types, like excitatory stellatecells and inhibitory interneurons, which are expected to have very weak LFP contributions [Lindénet al., 2011], but might still be measurable through their spiking activity. Similarly, spatially uniformlydistributed synaptic input to pyramidal neurons results in a negligible LFP contribution (Fig. 4C, timemarked by orange line), while the population might still contribute substantially to the MUA throughthe extracellular action potentials (Fig. 4E-F, time marked by orange line).
5. ECoG and EEG
In order to measure electric potentials in the immediate vicinity of neurons, like LFP and MUAsignals, we need to insert electrodes into the brain. This highly invasive technique is quite commonin animal studies, but can only be applied to humans when there is a clear medical need, for examplein patients with intractable epilepsy [Zangiabadi et al., 2019]. However, electric potentials generatedby neural activity extend beyond neural tissue and can also be measured outside the brain: Placingelectrodes on the brain surface, as in electrocorticography (ECoG), is a technique that requiressurgery. With electroencephalography (EEG), on the other hand, potentials are measured non-invasively, directly on top of the scalp.Since EEG electrodes are located relatively far away from the neuronal sources, the currentdipole approximation, eq. (12), combined with some head model, can be applied for computing EEGsignals [Nunez and Srinivasan, 2006; Ilmoniemi and Sarvas, 2019]. By collapsing the transmem-brane currents of a neuron simulation into one single current dipole moment, see eq. (13), we cancalculate EEG from arbitrary neural activity (Fig. 5). The current dipole approximation is howevernot unproblematic to use for computing ECoG signals, as the ECoG electrodes may be located tooclose to the signal sources for the approximation to apply, see Hagen et al. [2018].
Electric potentials measured on the scalp surface will be affected by the geometries and con-ductivities of the different constituents of the head [Nunez and Srinivasan, 2006]. This can be in-corporated in EEG calculations by applying simplified or more complex head models. A well-knownsimplified head model is the analytical four-sphere model, consisting of four concentric shells rep-resenting brain tissue, cerebrospinal fluid (CSF), skull and scalp, where the conductivity can be setindividually for each shell [Næss et al., 2017; Srinivasan et al., 1998; Nunez and Srinivasan, 2006](Fig. 6, Fig. 7A,B). More complex head models make use of high-resolution anatomical MRI-datato map out a geometrically detailed head volume conductor. The link between current dipoles inthe brain and resulting EEG signals is determined applying numerical methods such as the finiteelement method [Larson and Bengzon, 2013; Logg et al., 2012]. Once this link is established wecan in principle insert a dipole representing arbitrary neural activity into such a model, and compute14 igure 5:
EEG from apical synaptic input to population of pyramidal cells . A : The four-sphere head model withtwo orientations of the neural population from Fig. 4, either radial, mimicking a population in a gyrus (top) or tangential,mimicking a population in a sulcus (bottom). B : A snapshot of the EEG signal at the head surface for apical input (timemarked with blue dotted line in Fig. 4), for a radial population (top) or tangential population (bottom). The center of thepopulation is marked with a black dot. the resulting EEG signals quite straightforwardly. The New York Head model is an example of onesuch pre-solved complex head model, see Fig. 7C,D [Huang et al., 2016].The head models themselves introduce no essential frequency filtering of the EEG signal [Pfurtschellerand Cooper, 1975; Nunez and Srinivasan, 2006; Ranta et al., 2017], however, substantial spatial fil-tering will occur (Fig. 6). Additionally, the measured (or modeled) signals represent the averagepotential across the elecrode surface, and the large electrode sizes used in ECoG/EEG recordingscan have important effect on the measured signals [Nunez and Srinivasan, 2006; Hagen et al., 2018;Dubey and Ray, 2019].
6. Discussion
In the present chapter we have derived and applied well-established biophysical forward-modelingschemes for computing extracellular electric potentials recorded inside and outside the brain. Theseelectric potentials include spikes (both single-unit and multiunit activity (MUA)), LFP, ECoG and EEGsignals. The obvious application of this scheme is computation of electric signals from neuron andnetwork activity for comparison with experiments so that candidate models can be tested [Einevoll15 igure 6:
Effect of head inhomogeneities . The same current dipole will give substantially different potentials on the headsurface if the different conductivities of the head is included in a FEM model [Næss et al., 2017]. Left: Homogeneoussphere, with electrical conductivity, σ = 0 . S/m everywhere. Right: Standard four-sphere head model, with σ brain =0 . S/m, σ CSF = 5 σ brain , σ skull = σ brain / , σ scalp = σ brain . et al., 2019] or inferred [Goncalves et al., 2019; Skaar et al., 2020]. Another key application is thecomputation of benchmarking data for testing of data analysis methods such as spike sorting orCSD analysis [Denker et al., 2012].Inverse modeling of recorded electric potentials, that is, estimation of the neural sources un-derlying the signals, is inherently an ill-posed problem. This means that no unique solution for thesize and position of the sources exists. However, prior knowledge about the underlying sources andhow they generate the recorded signals, can be used to increase the identifiability. For example,several methods for the estimation of so-called current-source density (CSD) from LFP recordingshave been developed by building the present forward model into the CSD estimator [Pettersen et al.,2006; Potworowski et al., 2012; Cserpán et al., 2017].The present chapter has focused on the modeling of measurements of extracellular electricsignals. There are several other measurement modalities where detailed forward modeling could bepursued to allow for a more quantitative analysis of recorded data, such as magnetoencephalograpy(MEG), where magnetic fields are recorded outside the head, Voltage-sensitive dye imaging (VSDI),which reflects area-weighted neuronal membrane potentials [Chemla and Chavane, 2012], two-photon calcium imaging, which measures the intracellular calcium dynamics [Helmchen, 2012] andfunctional magnetic resonance imaging (fMRI), which reflects blood dynamics [Bartels et al., 2012],.While blood dynamics is typically not explicitly included in neural network models, MEG, VSDI andcalcium imaging are accessible through neuronal simulations of the type used to compute electricsignals. Similar to EEG, the MEG stems from the transmembrane currents of neurons and canbe computed based on the current dipoles of the underlying neurons [Hämäläinen et al., 1993;Ilmoniemi and Sarvas, 2019].The new version of our tool LFPy, which was used in generating theexamples in the present chapter, thus also includes the ability to compute MEG signals [Hagenet al., 2018]. 16 igure 7: The four-sphere head model and the NY Head model . EEG signals from population dipole resulting fromwaves of excitatory synaptic input to 10 000 layer 5 pyramidal cells from rat [Hay et al., 2011]. A : The four-spheremodel consisting of four concentrical shells: brain, CSF, skull and scalp. B : Maximum EEG signals ( φ ) on scalp surfaceelectrodes resulting from population dipole placed at location marked by orange star, computed with the four-spheremodel. C : Illustration of the New York Head model. D : EEG signals computed with the New York Head model, equivalentto panel B . Acknowledgements
This research has received funding from the European Union Horizon 2020 Framework Pro-gramme for Research and Innovation under Specific Grant Agreement No. 785907 and No. 945539[Human Brain Project (HBP) SGA2 and SGA3], and the Research Council of Norway (Notur, nn4661k;DigiBrain, no. 248828; INCF National Node, no. 269774).
References
Almeida, A.C.G., Texeira, H.Z., Duarte, M.A., Infantosi, A.F.C., 2004. Modeling extracellular spaceelectrodiffusion during lea/spl tilde/o’s spreading depression. IEEE Transactions on BiomedicalEngineering 51, 450–458. doi: .Bartels, A., Goense, J., Logothetis, N., 2012. Functional magnetic resonance imaging, in: Brette,R., Destexhe, A. (Eds.), Handbook of Neural Activity Measurement. Cambridge University Press,pp. 92–135. 17illeh, Y.N., Cai, B., Gratiy, S.L., Dai, K., Iyer, R., Gouwens, N.W., Abbasi-Asl, R., Jia, X.,Siegle, J.H., Olsen, S.R., Koch, C., Mihalas, S., Arkhipov, A., 2020. Systematic integration ofstructural and functional data into multi-scale models of mouse primary visual cortex. NeuronURL: , doi: , .Buccino, A.P., Einevoll, G.T., 2019. Mearec: a fast and customizable testbenchsimulator for ground-truth extracellular spiking activity. bioRxiv URL: , doi: , .Buccino, A.P., Kordovan, M., Ness, T.V., Merkt, B., Häfliger, P.D., Fyhn, M., Cauwenberghs, G.,Rotter, S., Einevoll, G.T., 2018. Combining biophysical modeling and deep learning for multi-electrode array neuron localization and classification. Journal of Neurophysiology 120, 1212–1232. URL: , doi: .Buccino, A.P., Kuchta, M., Jæger, K.H., Ness, T.V., Berthet, P., Mardal, K.A., Cauwenberghs, G.,Tveito, A., 2019. How does the presence of neural probes affect extracellular potentials? Journalof Neural Engineering 16. doi: .Buzsáki, G., Anastassiou, C.a., Koch, C., 2012. The origin of extracellular fields and currents–EEG,ECoG, LFP and spikes. Nature reviews. Neuroscience 13, 407–20. URL: , doi: .Cartailler, J., Kwon, T., Yuste, R., Holcman, D., 2018. Deconvolution of voltage sensor time seriesand electro-diffusion modeling reveal the role of spine geometry in controlling synaptic strength.Neuron 97, 1126–1136.Chemla, S., Chavane, F., 2012. Voltage-sensitive dye imaging, in: Brette, R., Destexhe, A. (Eds.),Handbook of Neural Activity Measurement. Cambridge University Press, pp. 92–135.Cserpán, D., Meszéna, D., Wittner, L., Tóth, K., Ulbert, I., Somogyvári, Z., Wójcik, D.K., 2017.Revealing the distribution of transmembrane currents along the dendritic tree of a neuron fromextracellular recordings. eLife 6. doi: .Delgado Ruz, I., Schultz, S.R., 2014. Localising and classifying neurons from high density MEArecordings. Journal of Neuroscience Methods 233, 115–128. URL: http://dx.doi.org/10.1016/j.jneumeth.2014.05.037 , doi: .Denker, M., Einevoll, G., Franke, F., Grün, S., Hagen, E., Kerr, J., Nawrot, M., Ness, T.B., Wójcik,T.W.D., 2012. Report from 1st INCF Workshop on Validation of Analysis Methods. TechnicalReport. International Neuroinformatics Coordinating Facility (INCF).Dubey, A., Ray, S., 2019. Cortical electrocorticogram (ecog) is a local signal. Journal of Neuro-science 39, 4299–4311. 18inevoll, G., Kayser, C., Logothetis, N., Panzeri, S., 2013a. Modelling and analysis of local fieldpotentials for studying the function of cortical circuits. Nature Reviews Neuroscience 14, 770–785.Einevoll, G.T., Destexhe, A., Diesmann, M., Grün, S., Jirsa, V., de Kamps, M., Migliore, M., Ness,T.V., Plesser, H.E., Schürmann, F., 2019. The Scientific Case for Brain Simulations. Neuron 102,735–744. doi: .Einevoll, G.T., Lindén, H., Tetzlaff, T., Ł ˛eski, S., Pettersen, K.H., 2013b. Local Field Potentials -Biophysical Origin and Analysis, in: Quiroga, R.Q., Panzeri, S. (Eds.), Principles of Neural Coding.CRC Press, Boca Raton, FL. chapter 3, pp. 37–60.Einevoll, G.T., Pettersen, K.H., Devor, A., Ulbert, I., Halgren, E., Dale, A.M., 2007. Laminar popu-lation analysis: estimating firing rates and evoked synaptic activity from multielectrode recordingsin rat barrel cortex. Journal of neurophysiology 97, 2174–90. URL: , doi: .Elbohouty, M., 2013. Electrical Conductivity of Brain Cortex Slices in Seizing and Non-seizing States.Ph.D. thesis. The University of Waikato.Ellingsrud, A.J., Solbrå, A., Einevoll, G.T., Halnes, G., Rognes, M.E., 2020. Finite element simula-tion of ionic electrodiffusion in cellular geometries. Frontiers in Neuroinformatics 14, 11. URL: , doi: .Frey, U., Egert, U., Heer, F., Hafizovic, S., Hierlemann, a., 2009. Microelectronic system for high-resolution mapping of extracellular electric fields applied to brain slices. Biosensors & bioelec-tronics 24, 2191–8. URL: , doi: .Gabriel, S., Lau, R.W., Gabriel, C., 1996. The dielectric properties of biological tissues: II. Measure-ments in the frequency range 10 Hz to 20 GHz. Physics in medicine and biology 41, 2251–69.URL: .Gardner, C.L., Jones, J.R., Baer, S.M., Crook, S.M., 2015. Drift-diffusion simulation of the ephapticeffect in the triad synapse of the retina. Journal of computational neuroscience 38, 129–42. URL: , doi: .Gł ˛abska, H., Potworowski, J., Ł ˛eski, S., Wójcik, D.K., 2014. Independent components of neuralactivity carry information on individual populations. PLoS One 9, e105071. URL: http://dx.doi.org/10.1371/journal.pone.0105071 , doi: .Gł ˛abska, H.T., Norheim, E., Devor, A., Dale, A.M., Einevoll, G.T., Wójcik, D.K., 2016. Generalizedlaminar population analysis (glpa) for interpretation of multielectrode data from cortex. Frontiersin neuroinformatics 10, 1. doi: .Gold, C., Henze, D.A., Koch, C., 2007. Using extracellular action potential recordings to constraincompartmental models. J. Comput. Neurosci. 23, 39–58. URL: http://dx.doi.org/10.1007/s10827-006-0018-2 , doi: .19oncalves, P.J., Lueckmann, J.M., Deistler, M., Nonnenmacher, M., Öcal, K., Bassetto, G., Chin-taluri, C., Podlaski, W.F., Haddad, S.A., Vogels, T.P., Greenberg, D.S., Macke, J.H., 2019. Trainingdeep neural density estimators to identify mechanistic models of neural dynamics. bioRxiv .Goto, T., Hatanaka, R., Ogawa, T., Sumiyoshi, A., Riera, J., Kawashima, R., 2010. An evaluation ofthe conductivity profile in the somatosensory barrel cortex of wistar rats. Journal of neurophysiol-ogy 104, 3388–3412.Gratiy, S.L., Halnes, G., Denman, D., Hawrylycz, M.J., Koch, C., Einevoll, G.T., Anastassiou, C.A.,2017. From Maxwell’s equations to the theory of current-source density analysis. EuropeanJournal of Neuroscience 45, 1013–1023. doi: .Grodzinsky, F., 2011. Fields, Forces, and Flows in Biological Systems. Garland Science, Taylor &Francis Group, London & New York.Hagen, E., Dahmen, D., Stavrinou, M.L., Lindén, H., Tetzlaff, T., Van Albada, S.J., Grün,S., Diesmann, M., Einevoll, G.T., 2016. Hybrid scheme for modeling local field potentialsfrom point-neuron networks. Cerebral Cortex 26, 4461–4496. doi: , arXiv:1511.01681 .Hagen, E., Næss, S., Ness, T.V., Einevoll, G.T., 2018. Multimodal modeling of neural networkactivity: computing LFP, ECoG, EEG and MEG signals with LFPy 2.0. Front Neuroinform 12.doi: .Hagen, E., Næss, S., Ness, T.V., Einevoll, G.T., 2019. LFPy – multimodal modeling of extracel-lular neuronal recordings in Python, in: Encyclopedia of Computational Neuroscience. Springer,New York, NY, p. 620286. URL: http://biorxiv.org/content/early/2019/05/03/620286.abstract , doi: .Haider, B., Schulz, D.P.A., Häusser, M., Carandini, M., 2016. Millisecond Coupling of Local FieldPotentials to Synaptic Currents in the Awake Visual Cortex. Neuron 90, 35–42. doi: .Halnes, G., Mäki-Marttunen, T., Keller, D., Pettersen, K.H., Andreassen, O.A., Einevoll, G.T., 2016.Effect of ionic diffusion on extracellular potentials in neural tissue. PLoS computational biology12, e1005193.Halnes, G., Mäki-Marttunen, T., Pettersen, K.H., Andreassen, O.A., Einevoll, G.T., 2017. Ion diffusionmay introduce spurious current sources in current-source density (CSD) analysis. Journal ofNeurophysiology 118, 114–120. URL: http://jn.physiology.org/lookup/doi/10.1152/jn.00976.2016 , doi: .Halnes, G., Ø stby, I., Pettersen, K.H., Omholt, S.W., Einevoll, G.T., 2015. An Electrodiffusive For-malism for Ion Concentration Dynamics in Excitable Cells and the Extracellular Space SurroundingThem, in: Advances in cognitive neurodynamics (IV). Springer Netherlands, pp. 353–360. URL: http://link.springer.com/chapter/10.1007/978-94-017-9548-7_50 .20alnes, G., Østby, I., Pettersen, K.H., Omholt, S.W., Einevoll, G.T., 2013. Electrodiffusive model forastrocytic and neuronal ion concentration dynamics. PLoS computational biology 9, e1003386.Hämäläinen, M., Hari, R., Ilmoniemi, R.J., Knuutila, J., Lounasmaa, O.V., 1993. Magnetoen-cephalography?theory, instrumentation, and applications to noninvasive studies of the workinghuman brain. Reviews of modern Physics 65, 413.Haufe, S., Huang, Y., Parra, L.C., 2015. A highly detailed FEM volume conductor model basedon the ICBM152 average head template for EEG source imaging and TCS targeting. Confer-ence proceedings : ... Annual International Conference of the IEEE Engineering in Medicine andBiology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference 2015,5744–5747. doi: .Hay, E., Hill, S., Schürmann, F., Markram, H., Segev, I., 2011. Models of neocorticallayer 5b pyramidal cells capturing a wide range of dendritic and perisomatic active proper-ties. PLoS Computational Biology 7, 1–18. URL: , doi: .Helmchen, F., 2012. Calcium imaging, in: Brette, R., Destexhe, A. (Eds.), Handbook of NeuralActivity Measurement. Cambridge University Press, pp. 92–135.Holcman, D., Yuste, R., 2015. The new nanophysiology: regulation of ionic flow in neuronal sub-compartments. Nature Reviews Neuroscience 16, 685–692.Holsheimer, J., 1987. Electrical conductivity of the hippocampal ca1 layers and application tocurrent-source-density analysis. Experimental brain research 67, 402–410.Holt, G., Koch, C., 1999. Electrical interactions via the extracellular potential near cell bodies. Journalof computational neuroscience 6, 169–184. URL: http://link.springer.com/article/10.1023/A:1008832702585 .Huang, Y., Parra, L.C., Haufe, S., 2016. The New York Head—A precise standardized vol-ume conductor model for EEG source localization and tES targeting. NeuroImage 140,150–162. URL: http://dx.doi.org/10.1016/j.neuroimage.2015.12.019 , doi: .Ilmoniemi, R.J., Sarvas, J., 2019. Brain Signals - Physics and Mathematics of MEG and EEG. MITPress, Cambridge, Massachusetts; London, England.Jackson, J.D., 1998. Classical electrodynamics. Third ed., Wiley.Joucla, S., Yvert, B., 2012. Modeling extracellular electrical neural stimulation: from basic under-standing to MEA-based applications. Journal of physiology, Paris 106, 146–58. URL: , doi: .Koch, C., 1999. Biophysics of computation: information processing in single neurons. 1st ed., OxfordUniversity Press: New York. 21arson, M.G., Bengzon, F., 2013. The finite element method: theory, implementation, and applica-tions. volume 10. Springer Science & Business Media.Łe¸ ski, S., Pettersen, K.H., Tunstall, B., Einevoll, G.T., Gigg, J., Wójcik, D.K., 2011. In-verse current source density method in two dimensions: inferring neural activation from mul-tielectrode recordings. Neuroinformatics 9, 401–25. URL: , doi: .Léonetti, M., Dubois-Violette, E., 1998. Theory of Electrodynamic Instabilities in BiologicalCells. Physical Review Letters 81, 1977–1980. URL: http://link.aps.org/doi/10.1103/PhysRevLett.81.1977 , doi: .Léonetti, M., Dubois-Violette, E., Homblé, F., 2004. Pattern formation of stationary transcellularionic currents in Fucus. Proceedings of the National Academy of Sciences of the United Statesof America 101, 10243–8. URL: , doi: .Łe¸ ski, S., Lindén, H., Tetzlaff, T., Pettersen, K.H., Einevoll, G.T., 2013. Frequency dependence ofsignal power and spatial reach of the local field potential. PLoS Comput Biol 9, e1003137.Lindén, H., Hagen, E., Łe¸ ski, S., Norheim, E.S., Pettersen, K.H., Einevoll, G.T., 2014. LFPy: a tool forbiophysical simulation of extracellular potentials generated by detailed model neurons. Frontiersin Neuroinformatics 7, 1–15. URL: http://journal.frontiersin.org/article/10.3389/fninf.2013.00041/abstract , doi: , arXiv:arXiv:1011.1669v3 .Lindén, H., Pettersen, K.H., Einevoll, G.T., 2010. Intrinsic dendritic filtering gives low-pass powerspectra of local field potentials. Journal of computational neuroscience 29, 423–44. URL: , doi: .Lindén, H., Tetzlaff, T., Potjans, T.C., Pettersen, K.H., Grün, S., Diesmann, M., Einevoll, G.T., 2011.Modeling the spatial reach of the LFP. Neuron 72, 859–72. URL: , doi: .Logg, A., Mardal, K.A., Wells, G., 2012. Automated solution of differential equations by the finiteelement method: The FEniCS book. volume 84. Springer Science & Business Media, Berlin,Heidelberg.Logothetis, N.K., Kayser, C., Oeltermann, A., 2007. In vivo measurement of cortical impedancespectrum in monkeys: implications for signal propagation. Neuron 55, 809–23. URL: , doi: .López-Aguado, L., Ibarz, J., Herreras, O., 2001. Activity-dependent changes of tissue resistivityin the ca1 region in vivo are layer-specific: modulation of evoked potentials. Neuroscience 108,249–262. 22opreore, C.L., Bartol, T.M., Coggan, J.S., Keller, D.X., Sosinsky, G.E., Ellisman, M.H., Se-jnowski, T.J., 2008. Computational modeling of three-dimensional electrodiffusion in bi-ological systems: application to the node of Ranvier. Biophysical journal 95, 2624–35. URL: , doi: .Lu, B., Zhou, Y.C., Huber, G.a., Bond, S.D., Holst, M.J., McCammon, J.A., 2007. Electrodiffusion: acontinuum modeling framework for biomolecular systems with realistic spatiotemporal resolution.The Journal of chemical physics 127, 135102. URL: , doi: .Luo, J., Macias, S., Ness, T.V., Einevoll, G.T., Zhang, K., Moss, C.F., 2018. Neural timing of stimulusevents with microsecond precision. PLoS biology 16, 1–22.Markram, H., Muller, E., Ramaswamy, S., Reimann, M.W., Abdellah, M., Sanchez, C.A., Ailamaki,A., Alonso-Nanclares, L., Antille, N., Arsever, S., et al., 2015. Reconstruction and simulation ofneocortical microcircuitry. Cell 163, 456–492.Martinsen, Ø.G., Grimnes, S., 2008. Bioimpedance and Bioelectricity Ba-sics. Academic Press; 2 edition. URL: .Mazzoni, A., Lindén, H., Cuntz, H., Lansner, A., Panzeri, S., Einevoll, G.T., 2015. Computing thelocal field potential (lfp) from integrate-and-fire network models. PLoS Comput Biol 11, e1004584.McIntyre, C.C., Grill, W.M., 2001. Finite Element Analysis of the Current-Density and Electric FieldGenerated by Metal Microelectrodes. Annals of Biomedical Engineering 29, 227–235. URL: , doi: .Mechler, F., Victor, J.D., 2012. Dipole characterization of single neurons from their extracellularaction potentials. Journal of computational neuroscience 32(1), 73–100. URL: , doi: .Miceli, S., Ness, T.V., Einevoll, G.T., Schubert, D., 2017. Impedance spectrum in cortical tissue: Im-plications for propagation of lfp signals on the microscopic level. eNeuro 4. URL: , doi: , .Mitzdorf, U., 1985. Current source-density method and application in cat cerebral cortex: investiga-tion of evoked potentials and eeg phenomena. Physiological reviews 65, 37–100.Moffitt, M.a., McIntyre, C.C., 2005. Model-based analysis of cortical recording with silicon microelec-trodes. Clinical neurophysiology 116, 2240–50. URL: , doi: .Mori, Y., 2009. From three-dimensional electrophysiology to the cable model: an asymp-totic study. arXiv preprint arXiv:0901.3914 , 1–39URL: http://arxiv.org/abs/0901.3914 , arXiv:arXiv:0901.3914v1 . 23ori, Y., Fishman, G.I., Peskin, C.S., 2008. Ephaptic conduction in a cardiac strand model with 3Delectrodiffusion. Proceedings of the National Academy of Sciences of the United States of Amer-ica 105, 6463–8. URL: , doi: .Mori, Y., Liu, C., Eisenberg, R.S., 2011. A Model of Electrodiffusion and Osmotic Water Flow and itsEnergetic Structure. arXiv preprint arXiv:1101.5193 arXiv:arXiv:1101.5193v1 .Mori, Y., Peskin, C., 2009. A numerical method for cellular electrophysiology based on the electrod-iffusion equations with internal boundary conditions at membranes. Communications in AppliedMathematics and Computational Science 4.1, 85–134. URL: http://msp.org/camcos/2009/4-1/p04.xhtml .Moulin, C., Glière, A., Barbier, D., Joucla, S., Yvert, B., Mailley, P., Guillemaud, R., 2008. A new3-D finite-element model based on thin-film approximation for microelectrode array recording ofextracellular action potential. IEEE transactions on bio-medical engineering 55, 683–92. URL: , doi: .Næss, S., Chintaluri, C., Ness, T.V., Dale, A.M., Einevoll, G.T., Wójcik, D.K., 2017. Cor-rected Four-Sphere Head Model for EEG Signals. Frontiers in Human Neuroscience 11,1–7. URL: http://journal.frontiersin.org/article/10.3389/fnhum.2017.00490/full ,doi: .Nanninga, P., 2008. A computational neuron model based on Poisson-Nernst-Planck theory.ANZIAM Journal 50, 46–59. URL: http://journal.austms.org.au/ojs/index.php/anziamj/article/view/1390 .Nelson, M.J., Pouget, P., 2010. Do electrode properties create a problem in interpreting local fieldpotential recordings? Journal of neurophysiology 103, 2315–7. URL: , doi: .Ness, T.V., Chintaluri, C., Potworowski, J., Łe¸ ski, S., Gła¸ bska, H., Wójcik, D.K., Einevoll, G.T., 2015.Modelling and Analysis of Electrical Potentials Recorded in Microelectrode Arrays (MEAs). Neu-roinformatics 13, 403–426. URL: http://link.springer.com/10.1007/s12021-015-9265-6 ,doi: .Ness, T.V., Remme, M.W.H., Einevoll, G.T., 2016. Active subthreshold dendritic conductancesshape the local field potential. Journal of Physiology 594, 3809–3825. doi: , arXiv:1512.04293 .Ness, T.V., Remme, M.W.H., Einevoll, G.T., 2018. h-Type Membrane Current Shapes the LocalField Potential from Populations of Pyramidal Neurons. Journal of Neuroscience 38, 6011–6024. URL: ,doi: .NeuroEnsamble, 2017. Elephant - electrophysiology analysis toolkit. URL: https://github.com/NeuralEnsemble/elephant . 24icholson, C., Freeman, J.A., 1975. Theory of current source-density analysis and determination ofconductivity tensor for anuran cerebellum. Journal of Neurophysiology 38, 356–368.Nicholson, C., Syková, E., 1998. Extracellular space structure revealed by diffusion analysis. Trendsin neurosciences 21, 207–15. URL: .Niederer, S., 2013. Regulation of ion gradients across myocardial ischemic border zones: a bio-physical modelling analysis. PloS one 8, e60323. URL: , doi: .Nunez, P.L., Srinivasan, R., 2006. Electric Fields of the Brain. Oxford University Press, New York.doi: , arXiv:arXiv:1011.1669v3 .Obien, M.E.J., Hierlemann, A., Frey, U., 2019. Accurate signal-source localization in brain slicesby means of high-density microelectrode arrays. Scientific Reports 9, 1–19. doi: .O’Connell, R., Mori, Y., 2016. Effects of glia in a triphasic continuum model of cortical spreadingdepression. Bulletin of Mathematical Biology 78, 1943–1967. URL: https://doi.org/10.1007/s11538-016-0206-9 , doi: .Pettersen, K.H., Devor, A., Ulbert, I., Dale, A.M., Einevoll, G.T., 2006. Current-source density es-timation based on inversion of electrostatic forward solution: effects of finite extent of neuronalactivity and conductivity discontinuities. Journal of neuroscience methods 154, 116–33. URL: , doi: .Pettersen, K.H., Einevoll, G.T., 2008. Amplitude variability and extracellular low-pass filtering ofneuronal spikes. Biophysical journal 94, 784–802. URL: , doi: .Pettersen, K.H., Hagen, E., Einevoll, G.T., 2008. Estimation of population firing rates and cur-rent source densities from laminar electrode recordings. Journal of computational neuro-science 24, 291–313. URL: , doi: .Pettersen, K.H., Lindén, H., Dale, A.M., Einevoll, G.T., 2012. Extracellular spikes and csd, in: Brette,R., Destexhe, A. (Eds.), Handbook of Neural Activity Measurement. Cambridge University Press,pp. 92–135. doi: .Pettersen, K.H., Lindén, H., Tetzlaff, T., Einevoll, G.T., 2014. Power laws from lin-ear neuronal cable theory: power spectral densities of the soma potential, soma mem-brane current and single-neuron contribution to the EEG. PLoS computational biology10, e1003928. URL: , doi: .25furtscheller, G., Cooper, R., 1975. Frequency dependence of the transmission of the EEG fromcortex to scalp. Electroencephalography and Clinical Neurophysiology 38, 93–96. doi: .Pods, J., 2017. A comparison of computational models for the extracellular potential of neurons.Journal of Integrative Neuroscience 16, 19–32. doi: , arXiv:1509.01481 .Pods, J., Schönke, J., Bastian, P., 2013. Electrodiffusion models of neurons and ex-tracellular space using the Poisson-Nernst-Planck equations–numerical simulation of theintra- and extracellular potential for an axon model. Biophysical journal 105, 242–54. URL: , doi: .Potworowski, J., Jakuczun, W., L ˛eski, S., Wójcik, D., 2012. Kernel current source density method.Neural computation 24, 541–75. URL: ,doi: .Ranck, J.B., 1963. Specific impedance of rabbit cerebral cortex. Experimental Neurology7, 144–152. URL: , doi: .Ranta, R., Le Cam, S., Tyvaert, L., Louis-Dorr, V., 2017. Assessing human brain impedance usingsimultaneous surface and intracerebral recordings. Neuroscience 343, 411–422. doi: .Reimann, M.W., Anastassiou, C.A., Perin, R., Hill, S.L., Markram, H., Koch, C., 2013. A biophys-ically detailed model of neocortical local field potentials predicts the critical role of active mem-brane currents. Neuron 79, 375–390. URL: http://linkinghub.elsevier.com/retrieve/pii/S0896627313004431 , doi: .Sætra, M.J., Einevoll, G.T., Halnes, G., 2020. An electrodiffusive, ion conserving pinsky-rinzel modelwith homeostatic mechanisms. bioRxiv .Savtchenko, L.P., Poo, M.M., Rusakov, D.A., 2017. Electrodiffusion phenomena in neuroscience: Aneglected companion. URL: http://dx.doi.org/10.1038/nrn.2017.101 , doi: .Schomburg, E.W., Anastassiou, C.A., Buzsaki, G., Koch, C., 2012. The Spiking Componentof Oscillatory Extracellular Potentials in the Rat Hippocampus. Journal of Neuroscience 32,11798–11811. URL: , doi: .Skaar, J.E.W., Stasik, A.J., Hagen, E., Ness, T.V., Einevoll, G.T., 2020. Estimation of neural networkmodel parameters from local field potentials (lfps). PLoS Computational Biology .Solbrå, A., Bergersen, A.W., van den Brink, J., Malthe-Sørenssen, A., Einevoll, G.T., Halnes, G.,2018. A Kirchhoff-Nernst-Planck framework for modeling large scale extracellular electrodiffusionsurrounding morphologically detailed neurons. PLoS Computational Biology 14, 1–26. doi: . 26rinivasan, R., Nunez, P.L., Silberstein, R.B., 1998. Spatial filtering and neocortical dynamics:estimates of eeg coherence. IEEE transactions on Biomedical Engineering 45, 814–826.Sterratt, D., Graham, B., Gillies, A., Willshaw, D., 2011. Principles of computational modelling inneuroscience. Cambridge University Press.Suzuki, M., Larkum, M.E., 2017. Dendritic calcium spikes are clearly detectable at thecortical surface. Nature Communications 8, 1–10. URL: http://dx.doi.org/10.1038/s41467-017-00282-4 , doi: .Tele ´nczuk, B., Dehghani, N., Le Van Quyen, M., Cash, S.S., Halgren, E., Hatsopoulos, N.G., Des-texhe, A., 2017. Local field potentials primarily reflect inhibitory neuron activity in human andmonkey cortex. Scientific reports 7, 40211.Tracey, B., Williams, M., 2011. Computationally efficient bioelectric field modeling and effectsof frequency-dependent tissue capacitance. Journal of Neural Engineering 8. doi: .Tuttle, A., Diaz, J.R., Mori, Y., 2019. A computational study on the role of glutamate and nmdareceptors on cortical spreading depression using a multidomain electrodiffusion model. PLoScomputational biology 15.Wagner, T., Eden, U., Rushmore, J., Russo, C.J., Dipietro, L., Fregni, F., Simon, S., Rot-man, S., Pitskel, N.B., Ramos-Estebanez, C., Pascual-Leone, A., Grodzinsky, A.J., Zahn, M.,Valero-Cabré, A., 2014. Impact of brain tissue filtering on neurostimulation fields: a modelingstudy. NeuroImage 85, 1048–57. URL: ,doi:10.1016/j.neuroimage.2013.06.079