Chapter: Estimation of Velocity Fields and Propagation on Non-Euclidian Domains: Application to the Exploration of Cortical Spatiotemporal DynamicsJulien Lefèvre, Sylvain Baillet[show abstract] [hide abstract]
ABSTRACT: Better understanding of the interrelationship between the brain’s structural architecture and functional processing is one of the leading questions in today’s integrative neuroscience. Non-invasive imaging techniques have revealed as major contributing tools to this endeavor, which obviously requires the cooperation of space and time-resolved experimental evidences. Electromagnetic brain mapping using magneto- and electro-encephalography (M/EEG) source estimation is so far the imaging method with the best trade-off between spatial and temporal resolution (∼1 cm and <1ms respectively, [4,5]). Combined with individual anatomical information from Magnetic Resonance Imaging (MRI) and statistical inference techniques , M/EEG source estimation has now reached considerable maturity and may indeed be considered as a true functional brain imaging technique. With or without considering the estimation of M/EEG generators as a priority, the analysis of M/EEG data is classically motivated by the detection of salient features in the time course of surface measures either/both at the sensor or/and cortical levels. These features of interest may be extracted from waveform peaks and/or their related time latencies, band-specific oscillatory patterns surging from a time-frequency decomposition of the data or regional activation blobs at the cortical level. By nature, the extraction of such features usually results from an extremely reductive – though pragmatic – point of view on the spatio-temporal dynamics of brain responses. It is pragmatic because it responds to a need for the reduction in the information mass from the original data. It is reductive though because most studies report on either/both the localization or/and the dynamical properties of brain events as defined according to the investigator, hence with an uncontrolled level of subjectivity.02/2013: pages 203-226;
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ABSTRACT: We report on mathematical methods for the exploration of spatiotemporal dynamics of Magneto- and Electro-Encephalography (MEG / EEG) surface data and/or of the corresponding brain activity at the cortical level, with high temporal resolution. In this regard, we describe how the framework and numerical computation of the optical flow--a classical tool for motion analysis in computer vision--can be extended to non-flat 2-dimensional surfaces such as the scalp and the cortical mantle. We prove the concept and mathematical well-posedness of such an extension through regularizing constraints on the estimated velocity field, and discuss the quantitative evaluation of the optical flow. The method is illustrated by simulations and analysis of brain image sequences from a ball-catching paradigm.Lecture Notes in Computer Science 02/2013; 20:470-81.
Article: Fiedler Vectors and Elongation of Graphs: A Threshold Phenomenon on a Particular Class of TreesJulien Lefèvre[show abstract] [hide abstract]
ABSTRACT: Let $G$ be a graph. Its laplacian matrix $L(G)$ is positive and we consider eigenvectors of its first non-null eigenvalue that are called Fiedler vector. They have been intensively used in spectral partitioning problems due to their good empirical properties. More recently Fiedler vectors have been also popularized in the computer graphics community to describe elongation of shapes. In more technical terms, authors have conjectured that extrema of Fiedler vectors can yield the diameter of a graph. In this work we present (FED) property for a graph $G$, i.e. the fact that diameter of a graph can be obtain by Fiedler vectors. We study in detail a parametric family of trees that gives indeed a counter example for the previous conjecture but reveals a threshold phenomenon for (FED) property. We end by an exhaustive enumeration of trees with at most 20 vertices for which (FED) is true and some perspectives.02/2013;
Article: Larger is twistier: Spectral analysis of gyrification (SPANGY) applied to adult brain size polymorphism.David Germanaud, Julien Lefèvre, Roberto Toro, Clara Fischer, Jessica Dubois, Lucie Hertz-Pannier, Jean-Francois Mangin[show abstract] [hide abstract]
ABSTRACT: The description of cortical folding pattern (CFP) is challenging because of geometric complexity and inter-subject variability. On a cortical surface mesh, curvature estimation provides a good scalar proxy of CFP. The oscillations of this function can be studied using a Fourier-like analysis to produce a power spectrum representative of the spatial frequency composition of CFP. First, we introduce an original method for the SPectral ANalysis of GYrication (Spangy), which performs a spectral decomposition of the mean curvature of the grey/white interface mesh based on the Laplace-Beltrami operator eigenfunctions. Spangy produces an ordered 7 bands power spectrum of curvature (B0-B6) and provides an anatomically relevant segmentation of CFP based on local spectral composition. A spatial frequency being associated with each eigenfunction, the bandwidth design assumes frequency doubling between consecutive spectral bands. Next, we observed that the last 3 spectral bands (B4, 5 and 6) accounted for 93% of the analyzed spectral power and were associated with fold-related variations of curvature, whereas the lower frequency bands were related to global brain shape. The spectral segmentation of CFP revealed 1st, 2nd and 3rd order elements associated with B4, B5 and B6 respectively. These elements could be related to developmentally-defined primary, secondary and tertiary folds. Finally, we used allometric scaling of frequency bands power and segmentation to analyze the relationship between the spectral composition of CFP and brain size in a large adult dataset. Total folding power followed a positive allometric scaling which did not divide up proportionally between the bands: B4 contribution was constant, B5 increased like total folding power and B6 much faster. Besides, apparition of new elements of pattern with increasing size only concerned the 3rd order. Hence, we demonstrate that large brains are twistier than smaller ones because of an increased number of high spatial frequency folds, ramifications and kinks that accommodate the allometric increase of cortical surface.NeuroImage 08/2012; 63(3):1257-72. · 5.89 Impact Factor
Guillaume Auzias, Julien Lefèvre, Arnaud Le Troter, Clara Fischer, Matthieu Perrot, Jean Régis, Olivier Coulon[show abstract] [hide abstract]
ABSTRACT: In the context of inter-subject brain surface matching, we present a parameterization of the cortical surface constrained by a model of cortical organization. The parameterization is defined via an harmonic mapping of each hemisphere surface to a rectangular planar domain that integrates a representation of the model. As opposed to previous conformal mapping methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso-coordinate axis in the model. Experiments on both hemispheres of 34 subjects are presented and results are very promising.Medical image computing and computer-assisted intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention. 01/2011; 14(Pt 2):310-7.