Informed spatial basis functions in minimum norm solutions for the electromagnetic source localisation problem.

Linear inverse solutions have been applied extensively to solve the bioelectromagnetic inverse problem. In contrast to discrete dipole equivalent models, linear inverse solutions do not require any assumptions about the number of active sources and lead to a fully 3D representation of the electrical activity of the brain. However the problem is underdetermined: there are many more parameters to estimate (relative to the number of dipole locations considered) than data available (relative to the number of electrodes). In order to ensure the uniqueness of the solution, existing methods generally apply constraints on the solution, for example: minimum 2-norm, maximum smoothness[1], optimal resolution[2], etc. These methods provide solutions with relatively poor spatial resolution because they neglect, wholly or in part, anatomical information relevant to the real source distribution.
Our method aims to model the spatial source distribution by using a set of basis functions. By appropriately defining these basis functions, we are able to include a priori information about the sources and our solutions will de facto belong to the subspace spanned by these basis functions. The priors enter as constraints of the covariance structure of the source power (over space), and are used to motivate the selection of a spatial basis set that maximises the information between the sources and their projection on that set. The orientation of each dipole is fixed and orthogonal to the cortical sheet, and therefore only the amplitude of the sources remains unknown. We then solve for the source distribution using two constraints: sources must correspond to dipoles localised to gray matter, and the 2D distribution of dipole strengths across the cortical surface must be spatially smooth.
In simulations conducted so far on noiseless instantaneous simulated data, we have obtained better resolution and more robust performance, even for deep sources, than could be achieved with other approaches [1,2]. Moreover the solutions are constrained to be anatomically realistic in orientation, amplitude and smoothness. We are planning to extend the approach to more realistic and noisy data, including a basis functions approach to constrain solutions in the temporal domain as well as in the spatial domain[3].
References:
1. Pascual-Marqui R.D., Michel C.M., Lehmann D. Low resolution electromagnetic tomography: a new method for localising electrical activity in the brain, Int. J. Psychol., 1994, 18:49-65.
2. Grave de Peralta Menendez R., Hauk O., Gonzalez Andino S., Vogt H., Michel C., Linear inverse solutions with optimal resolution kernels applied to electromagnetic tomography, Human Brain Mapping, 1997, 5:454-467.
3. Phillips C., Rugg M.D., Friston K.J., A priori spatio-temporal basis functions in minimum norm solutions, HBM99 abstract, to be published.