THE FAST MULTIPOLE METHOD FOR THE DIRECT E/MEG PROBLEM Maureen Clerc , Renaud Keriven , Olivier Faugeras , Jan Kybic , Th´ eo Papadopoulo Odyss´ ee project CERMICS Ecole Nationale des Ponts et Chauss´ ees 77455 Marne-La-Vall´ ee Cedex 2 France Odyss´ ee Project INRIA 2004 Route des Lucioles, BP93 06902 Sophia-Antipolis France ABSTRACT Reconstructing neuronal activity from MEG and EEG mea- surements requires the accurate calculation of the electro- magnetic field inside the head. The boundary element for- mulation of this problem leads to a dense linear system which is too large to be solved directly. We propose to accelerate the computations via the fast multipole method. This method approximates the electromagnetic interaction between surface elements by performing multipole expan- sions at a coarse resolution. It significantly reduces the com- putational complexity of the matrix-vector products needed for the iterative solution of the linear system, and avoids the storage of its matrix. We describe the single-level fast multi- pole method and present several experiments demonstrating its accuracy and performance. 1. INTRODUCTION Electro/magnetoencephalography (E/MEG) measures the small electric and magnetic fields on the surface of the skull [1]. The ultimate goal is to solve the inverse problem: reconstruct from these measurements the primary currents inside the cortex, due to brain activity. Its essential part is the solution of the forward problem: find the field caused by a known distribution of current sources. In this short article we focus on the computation of the electric potential since the magnetic field follows from the Biot-Savart law [1]. Anatomical data (MRI) can provide a head model [2] consisting of several regions of constant conductivity, representing for example the white matter, the cortex, the cerebrospinal fluid, the skull, and the scalp [3]. In the surface approach, the potential, sought at the boundaries between different regions, is governed by an integral equa- tion, which can be solved numerically with the boundary element method (BEM). Recent experiments indicate that existing BEM imple- mentations suffer from unacceptably large errors when the current source approaches the volume discontinuity [4]. Unfortunately, this is precisely the case in the brain, where the main primary current sources are supposedly the pyra- midal cells in the cortex, a layer only a few millimeters thick. Although the BEM is able to deal with complex sur- face geometry, realistic head models do not appear to offer greater accuracy than a simple concentric spheres model [5]. We believe that one of the reasons for this inadequacy is the simplicity of the surface model employed: the bound- aries are represented with at most several thousand triangles that cannot capture the fine cortex geometry nor the impor- tant spatial variations of the potential. Unfortunately, the computational complexity and storage requirements of the direct calculation of pairwise interactions between elements quickly become prohibitive. Iterative techniques [6] avoid the storage at the expense of increased number of opera- tions. The fast multipole method (FMM) is a standard tool in particle simulations and computational electromagnetics. It reduces the asymptotical complexity of the interaction eval- uation from , where is the number of parame- ters (degrees of freedom) in our model, ultimately down to , which allows to use models with an order of several magnitudes more elements than with the classical approach. 2. BOUNDARY ELEMENT METHOD 2.1. Surface equation The potential on a smooth surface separating two regions with conductivities , satisfies the following integral equation [4, 7]: for , (1)