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Edited by: Francesco Di Russo, Foro Italico University of Rome, Italy

Reviewed by: Zhimin Li, University of Wisconsin-Madison, United States; Moritz Dannhauer, Scientific Computing and Imaging Institute, United States; Ettore Ambrosini, Università degli Studi di Padova, Italy

*Correspondence: Gaute T. Einevoll

Daniel K. Wójcik

†These authors have contributed equally to this work.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The EEG signal is generated by electrical brain cell activity, often described in terms of current dipoles. By applying EEG forward models we can compute the contribution from such dipoles to the electrical potential recorded by EEG electrodes. Forward models are key both for generating understanding and intuition about the neural origin of EEG signals as well as inverse modeling, i.e., the estimation of the underlying dipole sources from recorded EEG signals. Different models of varying complexity and biological detail are used in the field. One such analytical model is the

Electroencephalography (EEG), that is, the recording of electrical potentials at the scalp, has been of key importance for probing human brain activity for more than half a century (Nunez and Srinivasan,

While the link between the current sources and the resulting potentials in principle is well described by volume-conductor theory, the practical application of this theory is not easy because the cortical tissue, the cerebrospinal fluid (CSF), the skull, and the scalp, all have different electrical conductivities (Nunez and Srinivasan,

Different forward modeling schemes approximate the geometries and conductivities of the head with various levels of biological detail. On one side we have the spherical head models that can provide analytical formulas for the EEG potentials generated by current dipoles. At the other side of the spectrum we have numerically comprehensive forward modeling schemes, including realistic geometries and electrical conductivities, even electrically anisotropic tissue (Bangera et al.,

In this paper, we address the four-sphere head model where the head is modeled as four concentric spherical layers. Here, the four layers represent brain tissue, CSF, skull, and scalp. The Poisson equation, which describes the electric fields of the brain within volume-conductor theory, is solved for each layer separately, and the mathematical solutions are matched at the layer interfaces to obtain an analytical expression for the EEG signal as set up by a current source in the brain tissue. The relatively small number of parameters makes the four-sphere model an obvious candidate for exploring and gaining intuition about the nature of EEG signals. Since the solution is analytical and requires little computation time compared to complex numerical schemes, it can be used to quickly test analysis methods and hypotheses. The most popular version of the four-sphere model was presented in Srinivasan et al. (

While conceptually clear, the mathematical expression of the four-sphere forward model is quite involved and rederiving the expression we discovered errors in the formulas both in the original paper and in the book. Due to the importance of the four-sphere model, we here derive and provide the correct analytical formulas for future reference. We tested our formulas by verifying that the solutions for neighboring layers matched on the layer boundaries. Moreover, when the conductivities for all the layers in the model were set to the same value, the model reduced to the well-known homogeneous single-sphere model as it should. We also verified that the model solution reduces to the formula for the extracellular potential from a current dipole in an infinite homogeneous space, when the layer radii go to infinity and the conductivities for all model layers are equal (not shown). As an application, we performed FEM simulations of the four-sphere model which were consistent with the corrected analytical formulas.

By assuming the quasi-static approximation of Maxwell's equations and using the well-established volume-conductor theory, the electric potential Φ can be found by solving the Poisson equation (Nunez and Srinivasan,

where _{s}(

Illustration of the four-sphere head model. _{z} from the center of the sphere. In all the subsequent figures, the dipole is placed in the _{z} = 7.8 cm).

Radii and electrical conductivities of the present four-sphere model.

1 | Brain | 7.9 | σ_{brain} = 0.33 |

2 | CSF | 8.0 | 5σ_{brain} |

3 | Skull | 8.5 | σ_{brain}/^{K} |

4 | Scalp | 9.0 | σ_{brain} |

The solution of Equation (1) takes different forms for tangential and radial dipoles, and any dipole can be decomposed into a linear combination of these two. The following derivations are based on Appendix G and H in Nunez and Srinivasan (

Nunez and Srinivasan (^{1}(^{s}(

Here, Φ^{s} is the extracellular potential measured at radius _{s}, from current dipole moment with magnitude _{z}. The conductivity of sphere _{s}, _{n}(cosθ) is the _{ij} ≡ σ_{i}/σ_{j} and _{ij} ≡ _{i}/_{j}:

Equations (5) and (6) are in accordance with Equations (G.1.9–10) in Appendix G of Nunez and Srinivasan (

The extracellular potential from a tangential dipole in a concentric-shells model is given by Equation (H.2.1) in Appendix H of Nunez and Srinivasan (

where φ is the azimuth angle and

In the results section we compare our analytical solution and the FEM simulations with the two published formulas for the potential in the four-sphere model given in Appendices G and H in Nunez and Srinivasan (_{1} was inserted in Equation (A-1), necessary to give potentials in units of volts. Secondly, a superscript in Equation (A-8) was changed, such that the right-hand-side included

To find the numerical solution of the four-sphere model we solved the Poisson equation (Equation (1)) using the FEM. The first step was to construct a 3D numerical mesh representing the four-sphere head model geometry. We used the open-source program

The dipole source was treated as two point current sources (Dirac δ functions) and the conductivity was set at each mesh point according to Table

We provide the

EEG potentials were computed on the scalp surface with the analytical four-sphere model Φ(_{4}, θ, ϕ) and compared with the results from the FEM simulations for a current dipole ^{−7} Am (two point sources of magnitude 100 μA separated by

EEG potentials computed with four-sphere model and FEM simulation for radial, tangential, and 45-degree dipole. _{z} = [0, 0, 7.8 cm] (red dot) and has a magnitude 10^{−7} Am to give scalp potentials some tens of microvolts in magnitude, typical for recorded EEG signals.

A more detailed comparison of EEG potentials predicted by the analytical model and the FEM model is shown in Figure _{z} and the measurement position vector

Analytical solution of four-sphere model matches FEM simulation. Scalp potentials from radial current dipole at position _{z} = 7.8 cm and magnitude 10^{−7} Am to give results in observable range, while still facilitating direct comparison with the original plots in Srinivasan et al. (_{skull}=σ_{brain}/20, σ_{brain}/40 and σ_{brain}/80, respectively.

As an additional control we tested the limiting case where the conductivity was set to be the same for all four shells, i.e., σ_{brain} = σ_{CSF} = σ_{skull} = σ_{scalp}, and equal to that of the brain (Table _{4}. For a dipole oriented along the radial direction inside a single homogeneous sphere, the surface potentials are given by Equation (6.7) in Nunez and Srinivasan (_{z}/_{4}. Comparison between the simplified four-sphere models and the homogeneous single-sphere model showed perfect agreement for the present formulation, while the formulas listed in Srinivasan et al. (

Analytical solution of the four-sphere model satisfies control test for limiting case. Four-sphere model in the limiting case where the conductivity of the skull, CSF, and scalp are equal to the conductivity of the brain, compared to the equivalent model for a single homogeneous sphere, Equation (19). We used a radial dipole of magnitude 10^{−7} Am positioned a distance _{z} = 7.8 cm away from the center of the sphere, consistent with Figures

In this note we have revisited the analytical four-sphere model for computing EEG potentials generated by current dipoles in the brain. The main contributions of this paper are the presentation of corrected and validated formulas, as well as the scripts for using them, allowing users to readily apply this important forward model in the field of EEG analysis.

In addition to facilitating the use of the four-sphere model in EEG signal analysis (see, e.g., Wong et al.,

Forward models with varying complexity are also used to test the accuracy of inverse methods which estimate the dipole source locations from the potentials and electrode positions. All inverse methods are based on a priori assumptions about the volume and conductivity of the brain. Their implementation requires a forward model encoded either as a lead field matrix or otherwise. The analytical solution of the four-sphere head model provides a way to quickly, yet exhaustively, obtain potentials for a wide range of dipole positions. This makes it an attractive option for testing the accuracy of inverse methods.

SN and TN derived the analytical expressions. CC developed the computational model and did the simulations. AD and GE designed the analytical study. DW designed the computational study. All authors wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Paul L. Nunez and Ramesh Srinivasan for useful personal contact. The study received funding from the Simula-UCSD-University of Oslo Research and PhD training (SUURPh) program, funded by the Norwegian Ministry of Education and Research, the European Union Horizon 2020 Research and Innovation Programme under Grant Agreement No. 720270 [Human Brain Project (HBP) SGA1], from the EC-FP7-PEOPLE sponsored NAMASEN Marie-Curie ITN grant 264872 and the Polish National Science Centre's OPUS grant (2015/17/B/ST7/04123).

The Supplementary Material for this article can be found online at: