## Volume Conduction Of Head Currents

Scalp potential may be expressed as a volume integral of dipole moment per unit volume over the entire brain provided P(r0, t) is defined generally rather than in columnar terms. For the important case of dominant cortical sources, scalp potential may be approximated by the following integral of dipole moment over the cortical volume 0:

If the volume element d0(r') is defined in terms of cortical columns, the volume integral may be reduced to an integral over the folded cortical surface. Equation (3) indicates that the time dependence of scalp potential is the weighted sum (or integral) of all dipole time variations in the brain, although deep dipole volumes typically make negligible contributions. The weighting function is called the vector Green's function, G(r, r'). It contains all geometric and conductive information about the head volume conductor. For the idealized case of sources in an infinite medium of scalar conductivity, s, the Green's function is

The vector G(r, r') is directed from each column (located at r') to scalp location r, as shown in Fig. 5. The numerator contains the vector difference between locations r and r'. The denominator contains the (scalar) magnitude of this same difference. The dot product of the two vectors in Eq. (3) indicates that only the dipole component along this direction contributes to scalp potential. In genuine heads, G(r, r') is much more complicated. The most common head models consist of three or four concentric spherical shells, representing brain, cerebrospinal fluid, skull, and scalp tissue with different electrical conductivities s. More complicated numerical methods may also be used to estimate G(r, r'), sometime employing MRI to determine tissue boundaries. The accuracy of both analytic and numerical methods is limited by incomplete knowledge of tissue conductivities. MRI has also been suggested as a future means of estimating tissue conductivities.

Despite these limitations preventing highly accurate estimates of the function G(r, r'), a variety of studies using concentric spheres or numerical methods have provided reasonable quantitative agreement with experiment. The cells generating scalp EEG are believed to have the following properties: First, in the case ofpotentials recorded without averaging, cells generating EEG are mostly close to the scalp surface. Potentials fall off with distance from source regions as demonstrated by Eq. (4). In genuine heads, tissue inhomogeneity (location-dependent properties) and anisotropy (direction-dependent properties) complicate this issue. For example, the low-conductivity skull tends to spread currents (and potentials) in directions tangent to its surface. Brain ventricles, the subskull cerebrospinal fluid layer, and skull holes (or local reductions in resisitance per unit area) may provide current shunting. Generally, however, sources closest to electrodes are expected to make the largest contributions to scalp potentials.

Second, the large pyramidal cells in cerebral cortex are aligned in parallel, perpendicular to local surface. This geometric arrangement encourages large extra-cranial electric fields due to linear superposition of contributions by individual current sources. Columnar sources P(r', t) aligned in parallel and synchronously active make the largest contribution to the scalp potential integral in Eq. (3). For example, a 1-cm2 crown of cortical gyrus contains about 110,000 minicolumns, approximately aligned. Over this small region, the angle between P(r', t) and G(r, r') in Eq. (3) exhibits relatively small changes. By "synchronous" sources, it is meant that the time dependence of P(r', t) is approximately consistent (phase locked) over the area in question. In this case, Eq. (3) implies that individual synchronous column sources add by linear superposition. In contrast, scalp potentials due to asynchronous sources are due only to statistical fluctuations—that is, imperfect cancellation of positive and negative contributions to the integral in Eq. (3). Scalp potential may be estimated as approximately proportional to the number of synchronous columns plus the square root of the number of asynchronous columns. For example, suppose 1% (s1»103) of the gyrial minicolumns produce synchronous sources P(r', t) and the other 99% of minicolumns (s2»105) produce sources with random time variations. The 1% synchronous minicolumn sources are expected to contribute approximately s1/ ^J~s2 or about three times as much to scalp potential measurements as the 99% random minicolumn sources.

Third, the observed ratio of brain surface (dura) potential magnitude to scalp potential magnitude for widespread cortical activity such as alpha rhythm is approximately in the 2-6 range. In contrast, this attenuation factor for very localized cortical epileptic spikes can be 100 or more. A general clinical observation is that a spike area of at least 6 cm2 of cortical surface must be synchronously active in order to be identified on the scalp. Such area contains about 700,000 minicolumns or 70 million neurons forming a dipole layer. These experimental observations are correctly predicted by Eq. (3).

Finally, for dipole layers partly in fissures and sulci, larger areas are required to produce measurable scalp potentials. First, the maximum scalp potential due to a cortical dipole oriented tangent to the scalp surface is estimated to be about one-third to one-fifth of the maximum scalp potential due to a dipole of the same strength and depth but orientated normal to the surface. Second, tangential dipoles tend to be located more in fissures and (deeper) sulci and may also tend to cancel due to opposing directions on opposite sides of the fissures and sulci. Third, and most important, synchronous dipole layers of sources with normal orientation covering multiple adjacent gyri can form, leading to large scalp potentials due to the product P(r', t) • G(r, r') having constant sign over the integral in Eq. (3).

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