Determining the inhomogeneity in Poisson's equation from incomplete Dirichlet boundary conditions: Applications in the neurosciences

Researchers and clinicians studying the electroencephalogram (EEG) and evoked potentials (EP) of the brain would like to localize and describe the neural generators of scalp-recorded potentials without recourse to invasive recording techniques such as depth electrodes or subdural patches. The choice of the underlying mathematical model is of critical importance to achieving this goal.; A mathematical model consisting of Poisson's equation with Dirichlet boundary conditions is effective in discriminating and localizing deep neural generators. However, the model consisting of Laplace's equation with Dirichlet boundary conditions is ineffective in localizing such sources.; The former model is tested using the Cortical Imaging Technique (CIT), which creates an artificial dipole layer in the brain that produces a potential field to match the empirical scalp-recorded electrode voltages. The theoretical field generated by CIT can be calculated anywhere in the conducting medium simulating the head.; The latter model is tested using the Spherical Harmonic Expansion (SHE) method. This method models the head with three conducting layers. However, this approach does not account for deep cerebral sources.; It has been shown that the calculation of radial current flow per unit area across the skull into the scalp yields information about the neural generators of scalp-recorded data not apparent from the scalp potential alone. We show that the calculation of a radial current flow at the level of the cortex complements information obtained from the cortical potential.; Two clinically interesting cases are given which demonstrate that CIT potential maps can give information not apparent from the scalp potential.; We develop data compression methods that can be used in the design of a publicly accessible archive of EEG signals. These compression techniques are based on a recently developed wavelet decomposition approach. Potential contour maps are compressed using only 20% of the wavelet coefficients without losing significant information. EEG spike data was found more compressible than seizure data with all five transforms tested. All transforms had equivalent performance when using 20-25% of the coefficients. The T-S transform outperformed the Daubechies wavelets and the Antonini wavelet when considering the full range of compression, 0-100%.