the potential varies linearly over each triangle (Gencer et al. 1999). Evaluating ro at each corner leads to a matrix equation for $ at the corners, which may be solved by inversion. Once $ is known on S, then (5.4) may be evaluated at any ro in V. Scalp potential values at the electrodes may be computed this way, or estimated using spline interpolation (see Sect. 8).

Aside from inhomogeneities and anisotropies ignored in spherical head models, the conductivity of head tissues are known within some (perhaps large) range of error (Foster and Schwan 1989). The brain conductivity o\ ~ 0.15 S/m (Stoy et al. 1982). The CSF conductivity a2 ~ 1.79 S/m (Baumann et al. 1997). The scalp conductivity a4 ~ 0.44 S/m (Geddes and Baker 1967). The conductivity of the living human skull, however, has been a source of mass confusion. Rush and Blanchard (1966) measured the conductivity ratio between the skull and that of saline in which the skull was immersed, and found conductivity ratios ranging from 50 to 300. Rush and Driscoll (1968) found a ratio near 80, then applied that ratio between the brain and skull, as though the living skull were saturated with brain-like rather than saline-like fluid. Most subsequent studies (e.g., Stok 1987) have used this ratio. Assuming the brain conductivity o\ ~ 0.15 S/m, for example, o\/a3 ~ 80 implies as ~ 0.002 S/m.

Since then evidence has accumulated that this early reasoning may greatly underestimate a3. Even within the context of the Rush and Driscoll (1968) study, assuming the saline conductivity a ~ 1.3 S/m implies a3 ~ 0.017 S/m. Kosterick et al. 1984 reported a3 ~ 0.012 S/m. Averaging the values reported in Law et al. (1993) suggests a3 ~ 0.018 S/m. Oostendorp et al.

(2000) reported <r3 — 0.015 S/m. This series of literature seems to implies consistently that <r3 — 0.015 S/m and a\/a3 — 10. This ratio is lower than the range 20-80 suggested by Nunez and Srinivasan (2005), due partly to a lower estimate of brain conductivity. With this skull conductivity, assuming the brain conductivity o\ — 0.33 S/m (Stok 1987), for example, gives the ratio <ti/<73 — 22. Early models assumed the brain and scalp conductivity were equal (Rush and Driscoll 1968). If this skull conductivity is compared to the scalp rather than the brain, a4/a3 — 29.

As discussed in Nunez and Srinivasan (2006), however, the effective conductivity of a single layered skull used in a volume conductor model may be substantially lower than its actual conductivity due to several shunting tissues not included in such models, e.g., extra CSF, the middle skull layer, and the anisotropic white matter. For example, consider a three-layered skull in which the inner skull layer conductivity is substantially higher than the inner and outer skull layers (as verified experimentally). Imagine a limiting case where the resistivity of the inner layer goes to zero so that no current enters the outer skull layer or scalp (zero scalp potential everywhere). The effective brain-to-skull conductivity ratio is infinite in this limiting case, even though the measured value based on a composite skull could easily be less than 20. This argument implies that effective brain-to-skull conductivity ratios cannot be accurately estimated from impedance measurements of composite skull tissue alone.

6 Data Acquisition

In EEG recordings, electric potential is measured on the scalp surface, and used to make inferences about brain activity. Although potentials relative to infinity are often considered in theoretical derivations, in practice only potential differences can be measured. Thus EEG measurements always involve the potential difference between two sites. This is accomplished using differential amplifiers, which include the measure electrodes, a reference electrode, and an "isolated common" electrode that takes the place of true ground.

Huhta and Webster (1973) presented an essentially complete analysis of electrocardiographic (ECG) recordings using differential amplifiers, including signal loss and 60 Hz noise. Several of their assumptions are either outdated or not applicable to EEG. First, they assumed that the subject was resistively coupled to earth ground. This simplification reduces the number of variables in the calculations, but is unsafe because it increases the risk of electric shock. It also allows more 60 Hz noise to enter the measurements because ambient electric potential fields in the recording environment exist relative to earth ground. Second, they assumed the grounding electrode was connected to the subjects foot, at maximal distance from the recording and reference electrodes which were located on the torso for cardiac recording. The thinking was that the foot would be electrically quiet, which may be true, but this increases 60 Hz noise because the entire body acts as an antenna.

Modern EEG systems are designed differently (Ferree et al. 2001). First, safety regulations require that the subject be isolated from ground so that contact with an electric source would not result a path to earth ground. This is accomplished by using an "isolated common" electrode that is electrically isolated from the ground of the power supply. In this configuration, the subject is only capacitively coupled to true ground, largely eliminating the risk of electric shock, and reducing 60 Hz noise. The measurement is then made as follows. The potential of both measurement and reference leads are taken relative to the common electrode, then their difference is amplified. Second, both the reference and common electrodes are located on the head in order to minimize 60 Hz common-mode noise sources, as well as physiological noise from cardiac sources.

The validity of the quasi-static approximation to Maxwell's equations in biological materials is equivalent to saying that the electric and magnetic fields propagate from the brain to the scalp instantaneously. In this sense, the temporal resolution of EEG (and MEG) is unlimited. Because most of the power in EEG time series falls below 100 Hz, typical sampling rates are 250 Hz, 500 Hz, and 1 kHz. Higher sampling rates are used to measure the brain-stem auditory evoked potential, and to adequately represent artifacts when EEG is recorded simultaneously with fMRI, but usually lower sampling rates are preferred because they result in smaller file sizes and faster analysis.

In digital signal processing, the Nyquist theorem states that power at frequency f in a signal must be sampled with interval At < 1/(2/). For fixed At, this means that only frequencies f < 1/(2At) are accurately represented; this is called the Nyquist frequency. Power at frequencies f > 1/(2At) are aliased, i.e., represented inaccurately as power at lower frequencies. To avoid this, EEG and other amplifiers sample in two stages. For a given choice of sampling rate At, analog filters are applied to remove signal power at frequencies f > 1/(2At), then the signal is sampled discretely. In this way, EEG amplifiers have a wide range of sampling rates that may be selected without aliasing.

In clinical EEG, speed, convenience, and culture typically dictate that only 19 electrodes be used, with inter-electrode spacing around 30 degrees. This configuration reveals large-scale brain activity reasonably well, and taking the potential difference between neighboring electrode pairs can isolate focal activity between those electrodes provided other conditions are met. Generally speaking, however, this low density misses much of the spatial information in the scalp potential. In research EEG, electrode arrays typically have 32, 64,

128, or 256 recording channels. The more electrodes, the more information, but there is a limit to the improvement.

The skull tends to smooth the scalp potential, compared to the brain surface or inner skull surface potential. Srinivasan et al. (1998) used spherical head models to quantify this effect. They generated random, many-dipole configurations in the cortex, and computed the scalp surface potentials. They sampled the scalp potential discretely using 19-, 32-, 64, and 128-channel arrays, and quantified the map differences for each array. They concluded that 128 electrodes are necessary to capture most of the spatial information available in the scalp potential, and that fewer than 64 channels can result in significant sampling errors. As in the time domain, if the scalp topography is sampled too sparsely, it suffers from aliasing artifacts. In the spatial domain, however, aliasing due to under-sampling can not be corrected by pre-filtering, as is done in the time domain.

This section describes a useful way of thinking about the spatial resolution of scalp EEG. Previous sections described how each dipole (specified by position and orientation) gives a unique scalp potential. In this way of thinking, the potential for a single dipole is normally computed at all electrodes. Alternatively, the same problem may be arranged so that the potential difference across a single electrode pair is computed for each dipole position and orientations. This yields the lead field vector L for each electrode pair, which may be computed from the electric field that would exist in the head if current were injected into those same electrodes. This seems less intuitive but, insofar as scalp measurements integrate over the activity of large cortical areas (10-100 cm2 ), this leads to a metric for the field of view of each electrode pair. The tabulation of the potential at every electrode, for each of a large but finite number of dipole locations and orientations in the brain, is called the lead field matrix. This quantity summarizes all the information about the head model, and is the starting point for inverse solutions.

Imagine that a single dipole is placed at a point rp inside a volume conductor, and oriented along the positive x-axis. Make no assumptions about the shape, homogeneity or isotropy of the volume conductor. Let A$ be the potential difference measured across two surface electrodes, and px be the dipole strength. Because Poisson's equation is linear in the sources, A$ must depend linearly upon the strength of the dipole, and this may be written algebraically as

where Lx is a proportionality constant. At the point rp, similar relations hold for the other two dipole orientations. If there were three perpendicular dipoles, one along each of three Cartesian axes, then would be the linear sum of each contribution.

where the last equality derives simply from the definition of vector dot product. The quantity L is called the lead field vector. Strictly speaking we have not shown that L behaves as a vector under coordinate transformations, but it must if its contraction with the vector p is to yield a scalar A$.

The reciprocity theorem (Helmholtz, 1853) gives an explicit expression for L. The mathematical techniques used in deriving it are similar to those used in boundary element modeling. Consider a conducting volume V bounded by a surface S. Make no assumptions about the shape or homogeneity of the volume conductor.5 Figure 4 shows two source and measurement configurations, denoted t1 and t2.

In configuration t1, the source is a dipole located in the volume and the measurement is made by surface electrodes at positions rA and rB. In configuration t2, the source is introduced "reciprocally" by injecting current through the surface electrodes, and the potential difference is considered across the dipole. Now use Green's theorem to relate the potential $ in one configuration to the current density J in the other. Consider the quantities

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