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. In this study, three well-established models (Buxton's, Friston's and Sotero's) are investigated. An exhaustive parameter sensitivity analysis (PSA) was conducted to study the marginal and joint influences of model parameters on the three main features of the BOLD response (namely the principal peak, the post-stimulus undershoot and the initial dip). In each model, parameters that have the greatest (and least) influence on the BOLD features as well as on the direction of variation of these features were identified. Among the three studied models, parameters were shown to affect the output features in different manners. Indeed, the main parameters revealed by the PSA were found to strongly depend on the way the flow(CBF)-metabolism(CMRO(2)) relationship is implemented (serial vs. parallel). This study confirmed that the model structure which accounts for the representation of the CBF-CMRO(2) relationship (oxygen supply by the flow vs. oxygen demand from neurons) plays a key role. More generally, this work provides substantial information about the tuning of parameters in the three considered models and about the subsequent interpretation of BOLD signals based on these models.

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. In this paper, we demonstrate how this constraint can be relaxed, allowing us to model more realistic head topologies. We describe the symmetric BEM for this new model. The symmetric BEM formulation uses both potentials and currents at the interfaces as unknowns and is in general more accurate than the alternative double-layer formulation.

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. In this paper we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single- and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms.

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). It creates a hierarchical structure of the elements and approximates far interactions using spherical harmonics expansions. The accelerated method is shown to be as accurate as the direct method, yet for large problems it is both faster and more economical in terms of memory consumption.

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.g. 4th order tensors) have been proposed instead. These, however, lack invariant features and computing them systematically is challenging. We present a simple and systematic method to compute a functionally complete set of invariants of a non-negative 3D 4th order tensor with respect to SO3. Intuitively, this transforms the tensor's non-unique ternary quartic (TQ) decomposition (from Hilbert's theorem) to a unique canonical representation independent of orientation - the invariants. The method consists of two steps. In the first, we reduce the 18 degrees-of-freedom (DOF) of a TQ representation by 3-DOFs via an orthogonal transformation. This transformation is designed to enhance a rotation-invariant property of choice of the 3D 4th order tensor. In the second, we further reduce 3-DOFs via a 3D rotation transformation of coordinates to arrive at a canonical set of invariants to SO3 of the tensor. The resulting invariants are, by construction, (i) functionally complete, (ii) functionally irreducible (if desired), (iii) computationally efficient and (iv) reversible (mappable to the TQ coefficients or shape); which is the novelty of our contribution in comparison to prior work. Results from synthetic and real data experiments validate the method and indicate its importance.

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.e. of finding which electrode corresponds to each point. This paper proposes a fast and robust solution to this latter problem based on combinatorial optimization. We design a specific energy that we minimize with a modified version of the Loopy Belief Propagation algorithm. Experiments on real data show that, with our method, a manual labeling of two or three electrodes only is sufficient to get the complete labeling of a 64 electrodes cap in less than 10 seconds.

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.e. of finding which electrode corresponds to each point. This paper proposes a fast and robust solution to this latter problem based on combinatorial optimization. We design a specific energy that we minimize with a modified version of the Loopy Belief Propagation algorithm. Experiments on real data show that, with our method, a manual labeling of two or three electrodes only is sufficient to get the complete labeling of a 64 electrodes cap in less than 10 seconds. However, it is shown to be robust to missing electrodes in the reconstructed data.

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. We present a method for automatically and accurately extracting surface meshes of several tissues of the head from anatomical magnetic resonance (MR) images. Anatomical connectivity can be extracted from diffusion tensor magnetic resonance images but, in the current state of the technology, it must be preceded by a robust estimation and regularization stage. We discuss our work based on variational principles and show how the results can be used to track fibers in the white matter (WM) as geodesics in some Riemannian space. We then go to the statistical modeling of functional magnetic resonance imaging (fMRI) signals from the viewpoint of their decomposition in a pseudo-deterministic and stochastic part that we then use to perform clustering of voxels in a way that is inspired by the theory of support vector machines and in a way that is grounded in information theory. Multimodal image matching is discussed next in the framework of image statistics and partial differential equations (PDEs) with an eye on registering fMRI to the anatomy. The paper ends with a discussion of a new theory of random shapes that may prove useful in building anatomical and functional atlases.