  We study the distribution of brain source from the most advanced brain
imaging technique, Magnetoencephalography (MEG), which measures the magnetic
fields outside the human head produced by the electrical activity inside the
brain. Common time-varying source localization methods assume the source
current with a time-varying structure and solve the MEG inverse problem by
mainly estimating the source moment parameters. These methods use the fact that
the magnetic fields linearly depend on the moment parameters of the source, and
work well under the linear dynamic system. However, magnetic fields are known
to be non-linearly related to the location parameters of the source. The
existing work on estimating the time-varying unknown location parameters is
limited. We are motivated to investigate the source distribution for the
location parameters based on a dynamic framework, where the posterior
distribution of the source is computed in a closed form discretely. The new
framework allows us not only to directly approximate the posterior distribution
of the source current, where sequential sampling methods may suffer from slow
convergence due to the large volume of measurement, but also to quantify the
source distribution at any time point from the entire set of measurements
reflecting the distribution of the source, rather than using only the
measurements up to the time point of interest. Both a dynamic procedure and a
switch procedure are proposed for the new discrete approach, balancing
estimation accuracy and computational efficiency when multiple sources are
present. In both simulation and real data, we illustrate that the new method is
able to provide comprehensive insight into the time evolution of the sources at
different stages of the MEG and EEG experiment.
