Four-sphere head model for EEG signals revisited

Electric potential recorded at the scalp (EEG) is dominated by contributions from current dipoles set by active neurons in the cortex. Estimation of these currents, called ’inverse modeling’, requires a ’forward’ model, which gives the potential when the positions, sizes, and directions of the current dipoles are known. Diﬀerent models of varying complexity and realism are used in the field. An important analytical example is the four-sphere model which assumes a four-layered spherical head where the layers represent brain tissue, cerebrospinal fluid (CSF), skull, and scalp, respectively. This model has been used extensively in the analysis of EEG recordings. Since it is analytical, it can also serve as a benchmark against which numerical schemes, such as the Finite Element Method (FEM), can be tested. While conceptually clear, the mathematical expression for the scalp potentials in the four-sphere model is quite cumbersome, and we observed the formulas presented in the literature to contain errors. We here derive and present the correct analytical formulas for future reference. They are compared with the results of FEM simulations of four-sphere model. We also provide scripts for computing EEG potentials in this model with the correct analytical formula and using FEM.


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Electroencephalography (EEG), that is, the recording of electrical potentials at the scalp, 23 has been of key importance for probing human brain activity for more than half a cen- where C(r, t) is the density of current sources. σ(r) is the position-dependent conductivity of the medium, here assumed to be isotropic so that σ(r) is a scalar. The four-sphere model is a specific solution of this equation which assumes that the conductive medium consists of four spherical layers representing specific constituents of the head: brain tissue, CSF, skull, and scalp ( Figure 1A). In the computations below, these layers are labeled by s = 1 to 4, respectively. The conductivity σ s (r) is assumed to be homogeneous, i.e., constant within each layer and independent of frequency (Pettersen et al., 2012).
In the examples below we assume the same values of conductivities and concentric shell radii as in Nunez and Srinivasan (2006), see Table 1. The solution of Equation (1) is subject to the following boundary conditions (where s = 1, 2, 3), assuring continuity of both electrical potential and current across the layer boundaries, and no current escaping the outer layer (Nunez and Srinivasan, 2006): ∂Φ 4 ∂r (r 4 ) = 0. potentials from a radial dipole in the four-sphere model: The potential in the inner sphere, 72 the brain, is given by Φ 1 (r, θ), while Φ s (r, θ) gives the potential in CSF, skull, and scalp, 73 for s = 2, 3, 4, respectively, Here, Φ s (r), is the extracellular potential measured at location r in shell number s, of 76 external radius r s , from current dipole moment p located at r z . The conductivity of sphere 77 Labels Name Radius (cm) σ (S/m) 1 Brain 7.9 0.33 2 CSF 8.0 5 σ brain 3 Skull 8.5 σ brain /K 4 Scalp 9.0 σ brain Table 1: Radii and electrical conductivities of the present four-sphere model. σ is the conductivity in each of the specified regions. Three variants of the model were considered with skull conductivity reduced by a factor K (20, 40, or 80) compared to the conductivity of the brain.
s is denoted by σ s , A s n and B s n are constants depending on the shell radii and conductivities, 78 and P n (cos θ) is the n-th Legendre Polynomial where θ is the angle between r and r z . From 79 the boundary conditions listed in Equations (2)-(4), we can compute A s n , for s = 1, 2, 3, 4 80 and B s n , for s = 2, 3, 4, using the notation σ ij ≡ σ i /σ j and r ij ≡ r i /r j : Equations (5) and (6)    1997). Figure 1B shows the resulting mesh corresponding to the set of radii listed in Table 1.

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Note that our 3D FEM model-geometry implementation consists of five spheres: scalp, 112 skull, cerebrospinal fluid (CSF), and two spheres together representing the brain tissue.

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However, the two innermost spheres (the innermost having a radius of 6 cm) are set to 114 have the same conductivity, i.e., the value for brain tissue listed in Table 1   As an additional control we tested the limiting case where the conductivity was set to be 154 the same for all four shells, i.e., σ brain = σ CSF = σ skull = σ scalp , and equal to that of the 155 Figure 2: EEG potentials computed with four-sphere model and FEM simulation for radial, tangential, and 45-degree dipole. (A) A radial current dipole placed in the brain in the model as described in Table 1. The dipole (black arrow) is located at r 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. (1 + f 2 − 2f cos θ) where f = r z /r 4 . Comparison between the simplified four-sphere models and the homoge-

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For the CSF Dirichlet boundary condition we can follow the same procedure as for the skull Dirichlet boundary condition, and we get, Inserting the expression for B 3 n from Equation (22): Here, we notice a typographical error in the expression for
Note that there's a subtle difference between the Y n presented here, and Nunez and second term of the numerator is a fraction. Here, the r n 23 factor should not be multiplied 250 by the whole fraction, but rather only the n n+1 -term in the numerator.
Inserting the expressions for A 2 n and B 2 n from Equations (25) and (24), we find,