The scalar simplification expressed in Eq. 3 and the applicability of Eq. 5 require a computational method as well as empirical validation. The general approach recognizes that intracranial fields must inevitably be sampled at discrete locations. This is illustrated in Fig. 1 (right) by the correspondence between the potential sampled by the recording electrode contacts (200 μm separation) and the continuous potential it represents. A slope (first derivative) can be computed for any adjacent pair of observations, and serve as an estimate for the curve between these points. Although the dipole that is illustrated is neither physiological nor adequately sampled, it is evident that the slope of the measured potential increases as the source is approached, inverting when it is passed. In contrast, a second derivative (i.e., change of slope) requires a minimum of three consecutive observations and serves as an estimate of the middle observation, but cannot be computed at the boundaries (0 and 1400 μm depths for the electrode shown in Fig. 1).
