Mathematical analysis of lead field expansions

The solution to the bioelectromagnetic inverse problem is discussed in terms of a generalized lead field expansion, extended to weights depending polynomially on the current strength. The expansion coefficients are obtained from the resulting system of equations which relate the lead field expansion to the data. The framework supports a family of algorithms which include the class of minimum norm solutions and those of the weighted minimum norm, including FOCUSS (FOCal Underdetermined System Solver), suitably modified to conform to the requirements of rotational invariance. The weighted minimum norm family is discussed in some detail, making explicit the dependence (or independence) of the weighting scheme on the modulus of the unknown current density vector. For all but the linear case, and with a single power in the weight, a highly nonlinear system of equations results. These equations are analyzed and their solution is reduced to tractable problems for a finite number of degrees of freedom. In the simplest magnetic field tomography (MFT) case, this is shown to possess expected properties for localized distributed sources. A sensitivity analysis supports this conclusion.