Soft Computation of Numerical Solutions to Differential Equations in EEG Analysis

Computational localization and modeling of functional activity within the brain, based on multichannel electroencephalographic (EEG) data are important in basic and clinical neuroscience. One of the key problems in analyzing EEG data is to evaluate surface potentials of a theoretical volume conductor model in response to an internally located current dipole with known parameters. Traditionally, this evaluation has been performed by means of either finite boundary or finite element methods which are computationally demanding. This paper presents a soft computing approach using an artificial neural network (ANN). Off-line training is performed for the ANN to map the forward solutions of the spherical head model to those of a class of spheroidal head models. When the ANN is placed on-line and a set of potential values of the spherical model are presented at the input, the ANN generalizes the knowledge learned during the training phase and produces the potentials of the selected spheroidal model with a desired eccentricity. In this work we investigate theoretical aspects of this soft-computing approach and show that the numerical computation can be formulated as a machining learning problem and implemented by a supervised function approximation ANN. We also show that, for the case of the Poisson equation, the solution is unique and continuous with respect to boundary surfaces. Our experiments demonstrate that this soft-computing approach produces highly accurate results with only a small fraction of the computational cost required by the traditional methods.