Surface Laplacian and fractal brain mapping

The electric potential recorded on the brain cortex results from the generation of sources of current. However, it depends on the reference electrode chosen and devices used. We propose here explicit formulae to compute the fractal dimension of experimental recordings by means of fractal interpolation. Another way of avoiding the reference dependence is to compute the scalp current density as surface Laplacian of the electroencephalogram. In practice, the value of the potential is known in a given number of electrodes, but the information about the density is not explicit. By using different interpolation methods by multivariate splines, formulae for the approximation of the density are obtained. These procedures are tested on a theoretical model of brain electrical potential given by the current of a single dipole inside the skull. By computing relative errors for different values of the eccentricity of the dipole it can be observed that errors decrease with increasing dipole eccentricities. The smallest errors are computed in the case of a pseudocubic spline. This method is used to perform two and three-dimensional brain mapping representations and to locate epileptic peaks.