A survey on integral equations for bioelectric modeling.

Bioelectric modeling problems, such as EEG, MEG, TES, DBS, and TMS, among others,
can be approached through the formulation and resolution of integral equations of the boundary element
method (BEM). Recently, it has been realized that the charge-based formulation of the BEM is naturally wellsuited for the application of the fast multipole method (FMM). The FMM is a powerful algorithm for the
computation of many-body interactions and is widely applied in electromagnetic modeling problems.
With the introduction of BEM-FMM in the context of bioelectromagnetism, the BEM can now compete
with the finite element method (FEM) in a number of application cases.

This survey has two goals: first, to give a modern account of the main BEM formulations in the
literature and their integration with FMM, directed to general researchers involved in development of
BEM software for bioelectromagnetic applications. Second, to survey different techniques and available
software, and to contrast different BEM and FEM approaches. As a new contribution, we showcase that
the charge-based formulation is dual to the more common surface potential formulation.