Complete set of invariants of a 4th order tensor: the 12 tasks of HARDI from ternary quartics.

Invariants play a crucial role in Diffusion MRI. In DTI (2nd order tensors), invariant scalars (FA, MD) have been successfully used in clinical applications. But DTI has limitations and HARDI models (e.

Relationship between flow and metabolism in BOLD signals: insights from biophysical models.

In many physiological or pathological situations, the interpretation of BOLD signals remains elusive as the intimate link between neuronal activity and subsequent flow/metabolic changes is not fully understood. During the past decades, a number of biophysical models of the neurovascular coupling have been proposed. It is now well-admitted that these models may bridge between observations (fMRI data) and underlying biophysical and (patho-)physiological mechanisms (related to flow and metabolism) by providing mechanistic explanations.

Combinatorial optimization for electrode labeling of EEG caps.

An important issue in electroencephalographiy (EEG) experiments is to measure accurately the three dimensional (3D) positions of the electrodes. We propose a system where these positions are automatically estimated from several images using computer vision techniques. Yet, only a set of undifferentiated points are recovered this way and remains the problem of labeling them, i.

Automatic labeling of EEG electrodes using combinatorial optimization.

An important issue in electroencephalography (EEG) experiments is to measure accurately the three dimensional (3D) positions of electrodes. We propose a system where these positions are automatically estimated from several images using computer vision techniques. Yet, only a set of undifferentiated points are recovered this way and remains the problem of labeling them, i.

Generalized head models for MEG/EEG: boundary element method beyond nested volumes.

Accurate geometrical models of the head are necessary for solving the forward and inverse problems of magneto- and electro-encephalography (MEG/EEG). Boundary element methods (BEMs) require a geometrical model describing the interfaces between different tissue types. Classically, head models with a nested volume topology have been used.

Fast multipole acceleration of the MEG/EEG boundary element method.

The accurate solution of the forward electrostatic problem is an essential first step before solving the inverse problem of magneto- and electroencephalography (MEG/EEG). The symmetric Galerkin boundary element method is accurate but cannot be used for very large problems because of its computational complexity and memory requirements. We describe a fast multipole-based acceleration for the symmetric boundary element method (BEM).

A common formalism for the integral formulations of the forward EEG problem.

The forward electroencephalography (EEG) problem involves finding a potential V from the Poisson equation inverted Delta x (sigma inverted Delta V) f, in which f represents electrical sources in the brain, and sigma the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the boundary element method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential.

Variational, geometric, and statistical methods for modeling brain anatomy and function.

We survey the recent activities of the Odyssée Laboratory in the area of the application of mathematics to the design of models for studying brain anatomy and function. We start with the problem of reconstructing sources in MEG and EEG, and discuss the variational approach we have developed for solving these inverse problems. This motivates the need for geometric models of the head.