[Best minimization to locate the brain sources of magnetic waves]

Locating the sources of brain magnetic fields is a basic question in magnetoencephalography (MEG). This question belongs to the indeterminate problem, that is, infinite solutions can be obtained from the equations of the measured small magnetic fields on the detector. Therefore a model of mathematics is needed to give more information. The major models of mathematics are the search method with many dipoles and the Lp minimum norm method. For the first model, if a global search is used, too much calculation time will be spent; if the gradient search is used, the searches usually converge to a local minimun. So other methods are introduced to find out the initial values, and then the gradient search can be used. For the second model, the depth of the brain sources will dependend on the choice of the weight functions which have several different types, and we do not know in advance which type is correct, so that it is difficult to have a good estimation of the depth of the brain sources. To overcome the difficulties, we use the method with three object functions which are the minimum Lp norm, the deviation between the calculated magnetic fields and the measured magnetic fields, and the number of the sources. This method is similar to the minimum norm method, a weight function is also introduced. But this weight function contains many reference variables which have to be determined through the minimization with three object functions. By choosing suitable weight function and its reference variables, the correct estimation of the depth of the brain sources can be achieved.