Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop

In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown that the same systems converge toward a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only. The considered class of equations is usually studied as a model of gene regulatory networks. It is not assumed that all decay rates are identical, which is biologically irrelevant, but has been done in the vast majority of previous studies. Our work relies on classical results about fixed points of monotone, concave operators acting on positive variables. Moreover, the used techniques are very likely to apply in more general contexts, opening directions for future work

Towards an efficient segmentation of small rodents brain: a short critical review

One of the most common tasks in small rodents MRI pipelines is the voxel-wise segmentation of the volume in multiple classes. While many segmentation schemes have been developed for the human brain, fewer are available for rodent MRI, often by adaptation from human neuroimaging. Common methods include atlas-based and clustering schemes. The former labels the target volume by registering one or more pre-labeled atlases using a deformable registration method, in which case the result depends on the quality of the reference volumes, the registration algorithm and the label fusion approach, if more than one atlas is employed. The latter is based on an expectation maximization procedure to maximize the variance between voxel categories, and is often combined with Markov Random Fields and the atlas based approach to include spatial information, priors, and improve the classification accuracy. Our primary goal is to critically review the state of the art of rat and mouse segmentation of neuro MRI volumes and compare the available literature on popular, readily and freely available MRI toolsets, including SPM, FSL and ANTs, when applied to this task in the context of common pre-processing steps. Furthermore, we will briefly address the emerging Deep Learning methods for the segmentation of medical imaging, and the perspectives for applications to small rodents

Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing and Proximity Analysis

In the past decade there has been a resurgence of interest in nonlinear dimension reduction. Among new proposals are “Local Linear Embedding,” “Isomap,” and Kernel Principal Components Analysis which all construct global low-dimensional embeddings from local affine or metric information. We introduce a competing method called “Local Multidimensional Scaling” (LMDS). Like LLE, Isomap, and KPCA, LMDS constructs its global embedding from local information, but it uses instead a combination of MDS and “force-directed” graph drawing. We apply the force paradigm to create localized versions of MDS stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces. We solve the problem of tuning parameter selection with a meta-criterion that measures how well the sets of K-nearest neighbors agree between the data and the embedding. Tuned LMDS seems to be able to outperform MDS, PCA, LLE, Isomap, and KPCA, as illustrated with two well-known image datasets. The meta-criterion can also be used in a pointwise version as a diagnostic tool for measuring the local adequacy of embeddings and thereby detect local problems in dimension reductions