F. Carbonell

H. de la Cruz, R. J. Biscay, J. C. Jimenez et al.

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge-Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.

F. Carbonell, Y. Iturria-Medina, J. C. Jimenez

The combination of the multiple shooting strategy with the generalized Gauss-Newton algorithm turns out in a recognized method for estimating parameters in ordinary differential equations (ODEs) from noisy discrete observations. A key issue for an efficient implementation of this method is the accurate integration of the ODE and the evaluation of the derivatives involved in the optimization algorithm. In this paper, we study the feasibility of the Local Linearization (LL) approach for the simultaneous numerical integration of the ODE and the evaluation of such derivatives. This integration approach results in a stable method for the accurate approximation of the derivatives with no more computational cost than the that involved in the integration of the ODE. The numerical simulations show that the proposed Multiple Shooting-Local Linearization method recovers the true parameters value under different scenarios of noisy data.

P. Bellec, Y. Benhajali, F. Carbonell et al.

Alterations in brain connectivity have been associated with a variety of clinical disorders using functional magnetic resonance imaging (fMRI). We investigated empirically how the number of brain parcels (or scale) impacted the results of a mass univariate general linear model (GLM) on connectomes. The brain parcels used as nodes in the connectome analysis were functionnally defined by a group cluster analysis. We first validated that a classic Benjamini-Hochberg procedure with parametric GLM tests did control appropriately the false-discovery rate (FDR) at a given scale. We then observed on realistic simulations that there was no substantial inflation of the FDR across scales, as long as the FDR was controlled independently within each scale, and the presence of true associations could be established using an omnibus permutation test combining all scales. Second, we observed both on simulations and on three real resting-state fMRI datasets (schizophrenia, congenital blindness, motor practice) that the rate of discovery varied markedly as a function of scales, and was relatively higher for low scales, below 25. Despite the differences in discovery rate, the statistical maps derived at different scales were generally very consistent in the three real datasets. Some seeds still showed effects better observed around 50, illustrating the potential benefits of multiscale analysis. On real data, the statistical maps agreed well with the existing literature. Overall, our results support that the multiscale GLM connectome analysis with FDR is statistically valid and can capture biologically meaningful effects in a variety of experimental conditions.