Julien Lefèvre

A Reaction-Diffusion Model of Human Brain Development

Cortical folding exhibits both reproducibility and variability in the geometry and topology of its patterns. These two properties are obviously the result of the brain development that goes through local cellular and molecular interactions which have important consequences on the global shape of the cortex. Hypotheses to explain the convoluted aspect of the brain are still intensively debated and do not focus necessarily on the variability of folds. Here we propose a phenomenological model based on reaction-diffusion mechanisms involving Turing morphogens that are responsible for the differential growth of two types of areas, sulci (bottom of folds) and gyri (top of folds). We use a finite element approach of our model that is able to compute the evolution of morphogens on any kind of surface and to deform it through an iterative process. Our model mimics the progressive folding of the cortical surface along foetal development. Moreover it reveals patterns of reproducibility when we look at several realizations of the model from a noisy initial condition. However this reproducibility must be tempered by the fact that a same fold engendered by the model can have different topological properties, in one or several parts. These two results on the reproducibility and variability of the model echo the sulcal roots theory that postulates the existence of anatomical entities around which the folding organizes itself. These sulcal roots would correspond to initial conditions in our model. Last but not least, the parameters of our model are able to produce different kinds of patterns that can be linked to developmental pathologies such as polymicrogyria and lissencephaly. The main significance of our model is that it proposes a first approach to the issue of reproducibility and variability of the cortical folding.

Larger is twistier: spectral analysis of gyrification (SPANGY) applied to adult brain size polymorphism.

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.

Model-Driven Harmonic Parameterization of the Cortical Surface: HIP-HOP

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 landmark-based registration methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso-coordinate axis in the model. This strategy overcomes some limitation to sulcus-based registration techniques such as topological variability in sulcal landmarks across subjects. Experiments on 62 subjects with manually traced sulci are presented and compared with the result of the Freesurfer software. The evaluation involves a measure of dispersion of sulci with both angular and area distortions. We show that the model-based strategy can lead to a natural, efficient and very fast (less than 5 min per hemisphere) method for defining inter subjects correspondences. We discuss how this approach also reduces the problems inherent to anatomically defined landmarks and open the way to the investigation of cortical organization through the notion of orientation and alignment of structures across the cortex.

Are Developmental Trajectories of Cortical Folding Comparable Between Cross-sectional Datasets of Fetuses and Preterm Newborns?

Magnetic resonance imaging has proved to be suitable and efficient for in vivo investigation of the early process of brain gyrification in fetuses and preterm newborns but the question remains as to whether cortical-related measurements derived from both cases are comparable or not. Indeed, the developmental folding trajectories drawn up from both populations have not been compared so far, neither from cross-sectional nor from longitudinal datasets. The present study aimed to compare features of cortical folding between healthy fetuses and early imaged preterm newborns on a cross-sectional basis, over a developmental period critical for the folding process (21-36 weeks of gestational age [GA]). A particular attention was carried out to reduce the methodological biases between the 2 populations. To provide an accurate group comparison, several global parameters characterizing the cortical morphometry were derived. In both groups, those metrics provided good proxies for the dramatic brain growth and cortical folding over this developmental period. Except for the cortical volume and the rate of sulci appearance, they depicted different trajectories in both groups suggesting that the transition from into ex utero has a visible impact on cortical morphology that is at least dependent on the GA at birth in preterm newborns.

Optical Flow and Advection on 2-Riemannian Manifolds: A Common Framework

Dynamic pattern analysis and motion extraction can be efficiently addressed using optical flow techniques. This paper presents a generalization of these questions to nonflat surfaces, where optical flow is tackled through the problem of evolution processes on non-euclidean domains. The classical equations of optical flow in the euclidean case are transposed to the theoretical framework of differential geometry. We adopt this formulation for the regularized optical flow problem, prove its mathematical well posedness, and combine it with the advection equation. The optical flow and advection problems are dual: A motion field may be retrieved from some scalar evolution using optical flow; conversely, a scalar field may be deduced from a velocity field using advection. These principles are illustrated with qualitative and quantitative evaluations from numerical simulations bridging both approaches. The proof-of-concept is further demonstrated with preliminary results from time-resolved functional brain imaging data, where organized propagations of cortical activation patterns are evidenced using our approach.

The birth of musical emotion: a depth electrode case study in a human subject with epilepsy.

Intracranial electroencephalography was recorded in an epileptic patient when he was listening to dissonant and consonant chords and to minor and major chords. Changes in dissonance induced event‐related potentials (ERPs) in the auditory areas from 200 ms onward, in the orbito‐frontal cortex (500–1000 ms), and later in the amygdala and anterior cingulate gyrus (1200–1400 ms), suggesting the sequential involvement of these brain structures in implicit emotional judgment of musical dissonance. Changes in musical mode induced ERPs only in the orbito‐frontal cortex (500–1000 ms), emphasizing the implication of this frontal region in emotional judgment of pleasant music.

Fast surface-based measurements using first eigenfunction of the Laplace-Beltrami Operator: Interest for sulcal description

In this paper we propose a fast method to compute the longitudinal extension of surfaces using the extrema of the first eigenfunction of Laplace-Beltrami Operator and the hot spots conjecture. We also propose an original definition of the surface width based on the distance to the longest geodesic. We show that the implementation of our new definition of length is consistent with the one computed from brute force and that the time complexity is considerably improved. We have tested the numerical efficiency of our approach on simple simulations and applied it to cortical surface patches from a real MRI dataset. Besides our approach enriches global descriptors of sulci shapes with a third dimension : length, depth and now width.

Model-driven harmonic parameterization of the cortical surface

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 landmark-based registration methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso-coordinate axis in the model. This strategy overcomes some limitation to sulcus-based registration techniques such as topological variability in sulcal landmarks across subjects. Experiments on 62 subjects with manually traced sulci are presented and compared with the result of the Freesurfer software. The evaluation involves a measure of dispersion of sulci with both angular and area distortions. We show that the model-based strategy can lead to a natural, efficient and very fast (less than 5 min per hemisphere) method for defining inter subjects correspondences. We discuss how this approach also reduces the problems inherent to anatomically defined landmarks and open the way to the investigation of cortical organization through the notion of orientation and alignment of structures across the cortex.

A reaction-diffusion model of the human brain development

The anatomical variability of the human brain folds remains an unclear and challenging issue. Several hypotheses coexist for explaining the rapid development of cortical sulci and it is clear that understanding their variability would improve the comparison of anatomical and functional data across cohorts of subjects. In this article we propose to extend a model of cortical folding based on reaction-diffusion mechanisms. The originality of our approach lies in the fact that the surface on which these mechanisms take place is deformed iteratively and engenders geometric patterns that can be linked to cortical sulci. We show that some statistic properties of our model can reflect the variability of sulcal structures.

Quasi-isometric length parameterization of cortical sulci: Application to handedness and the central sulcus morphology

We present in this paper a method to perform a length parameterization of cortical sulcus meshes. Such parameterization allows morphological features to be localized in a normalized way along the length of the sulcus and can be used to perform population studies and group comparisons. Our method uses the second eigenfunction of the Laplace-Beltrami operator, and the resulting parameterization is quasi-isometric. The process is validated on the central sulci of a set of subjects and its efficiency is demonstrated by quantifying morphological differences between left and right-handed subjects.