Mathematical analysis of lead field expansions

The solution to the bioelectromagnetic inverse problem is discussed in term s of a generalized lead field expansion, extended to weights depending poly nomially on the current strength. The expansion coefficients are obtained f rom the resulting system of equations which relate the lead field expansion to the data. The framework supports a family of algorithms which include t he class of minimum norm solutions and those of weighted minimum norm, incl uding FOCUSS (suitably modified to conform to requirements of rotational in variance), The weighted-minimum-norm family is discussed in some detail, ma king explicit the dependence (or Independence) of the weighting scheme on t he modulus of the unknown current density vector. For all but the linear ca se, and with a single power in the weight, a highly nonlinear system of equ ations results. These are analyzed and their solution reduced to tractable problems for a finite number of degrees of freedom. In the simplest magneti c field tomography (MFT) case, this is shown to possess expected properties for localized distributed sources. A sensitivity analysis supports this co nclusion.