Infinitesimal Lyapunov functions for singular flows

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative DX of the vector field X together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.Comment: 37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2) of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the statement of this item was false; corrected items (1) and (2) of Theorem 2.23 and their proofs. Included Example 6 on smooth reduction of families of quadratic forms. The published version in Math Z journal needs an errat

Longitudinal In Vivo Imaging of Retinal Ganglion Cells and Retinal Thickness Changes Following Optic Nerve Injury in Mice

Retinal ganglion cells (RGCs) die in sight-threatening eye diseases. Imaging RGCs in humans is not currently possible and proof of principle in experimental models is fundamental for future development. Our objective was to quantify RGC density and retinal thickness following optic nerve transection in transgenic mice expressing cyan fluorescent protein (CFP) under control of the Thy1 promoter, expressed by RGCs and other neurons.A modified confocal scanning laser ophthalmoscopy (CSLO)/spectral-domain optical coherence tomography (SD-OCT) camera was used to image and quantify CFP+ cells in mice from the B6.Cg-Tg(Thy1-CFP)23Jrs/J line. SD-OCT circle (1 B-scan), raster (37 B-scans) and radial (24 B-scans) scans of the retina were also obtained. CSLO was performed at baseline (n = 11) and 3 (n = 11), 5 (n = 4), 7 (n = 10), 10 (n = 6), 14 (n = 7) and 21 (n = 5) days post-transection, while SD-OCT was performed at baseline and 7, 14 and 35 days (n = 9) post-transection. Longitudinal change in CFP+ cell density and retinal thickness were computed. Compared to baseline, the mean (SD) percentage CFP+ cells remaining at 3, 5, 7, 10, 14 and 21 days post-transection was 86 (9)%, 63 (11)%, 45 (11)%, 31 (9)%, 20 (9)% and 8 (4)%, respectively. Compared to baseline, the mean (SD) retinal thickness at 7 days post-transection was 97 (3)%, 98 (2)% and 97 (4)% for the circle, raster and radial scans, respectively. The corresponding figures at 14 and 35 days post-transection were 96 (3)%, 97 (2)% and 95 (3)%; and 93 (3)%, 94 (3)% and 92 (3)%.Longitudinal imaging showed an exponential decline in CFP+ cell density and a small (≤8%) reduction in SD-OCT measured retinal thickness post-transection. SD-OCT is a promising tool for detecting structural changes in experimental optic neuropathy. These results represent an important step towards translation for clinical use