Physiological Foundations of Quantitative EEG Analysis - Quantitative EEG Analysis Methods and Clinical Applications - page 11

Relation (1.8) provides an estimate of reference-free potential in terms of
recorded potentials. Because we cannot measure the potentials over an entire closed
surface of an attached head, the first term on the right side of (1.7) will not generally
vanish. Due to sparse spatial sampling, the average reference is expected to provide
a very poor approximation if applied with the standard 10-20 electrode system. As
the number of electrodes increases, the error in approximation (1.8) is expected to
decrease. Like any other reference, the average reference provides biased estimates
of reference-independent potentials. Nevertheless, when used in studies with large
numbers of electrodes (say, 100 or more), it often provides a plausible estimate of
reference-independent potentials [12]. Because the reference issue is critical to EEG
interpretation, transformation to the average reference is often appropriate before
application of other transformations, as discussed in later chapters.

1.7 The Surface Laplacian

The process of relating recorded scalp potentials V ( r i , r R , t ) to the underlying brain
mesosource function P ( r , t ) has long been hampered by: (1) reference electrode dis-
tortions and (2) inhomogeneous current spreading by the head volume conductor.
The average reference method discussed in Section 1.6 provides only a limited solu-
tion to problem 1 and fails to address problem 2 altogether. By contrast, the surface
Laplacian completely eliminates problem 1 and provides a limited solution to prob-
lem 2. The surface Laplacian is defined in terms of two surface tangential coordi-
nates, for example, spherical coordinates (

) or local Cartesian coordinates ( x , y ).
From (1.6), with the understanding that the reference potential is spatially constant,
we obtain the surface Laplacian in terms of (any) reference potential:

θ , φ

( ) ( )

2 2

∂ Vx y ,, , r t ∂ Vx y , , r , t

() ( )

i i R i i R

L ≡∇

2

Φ r , t =∇

2

V r , r , t = + (1.9)

Si S i S i R

2 2

∂ x ∂ y

i i

The physical basis for relating the scalp surface Laplacian to the dura (or inner
skull) surface potential is based on Ohm's law and the assumption that skull con-
ductivity is much lower than that of contiguous tissue (by at least a factor of 5 or
so). In this case most of the source current that reaches the scalp flows normal to the
skull. With this approximation, the following approximate expression for the sur-
face Laplacian is obtained in terms of the local outer skull potential Φ Ki and inner

skull (outer CSF) potential Φ Ci [2]:

( )

LA
Si

≈ ΦΦ (1.10)

i Ki Ci

The parameter A i depends on several tissues thicknesses and conductivities, which
are assumed constant over the surface to first approximation. Simulations indicate
minimal falloff of potential through the scalp so that Φ K reasonably approximates

scalp surface potential.
Interpretation of L S depends critically on the nature of the sources. When corti-
cal sources consist of large dipole layers, the potential falloff through the skull is
minimal so Φ K Φ C and the surface Laplacian is very small. By contrast, when corti-