A Newton Algorithm for the Nearest Correlation Matrix

Firstly, we describe and investigate the algorithm of Qi and Sun which solves the problem of finding the nearest correlation matrix to a symmetric matrix. This algorithm claims a quadratic convergence. We discuss improving this algorithm's efficiency and reliability and detect a problem when we are aiming at a nearest correlation matrix with a high accuracy, using small error tolerences. As a consequence, we suggest a modified version, based on the algorithm of Qi and Sun, which is also a quadratically convergent algorithm, has improved efficiency and is modified so that the algorithm can return the nearest correlation matrix to high accuracy showing a robust and reliable behaviour. Secondly, we investigate the general alternating projections method and also Higham's alternating projections method for the nearest correlation matrix. We discuss variations of the latter and include a further projection which allows more constraints to be added to the problem. We introduce a new algorithm and compare its convergence behaviour with Higham's alternating projections method

Repeated hapten exposure induces persistent tactile sensitivity in mice modeling localized provoked vulvodynia - Fig 1

<p><b>Timeline of oxazolone sensitization, challenge, and post-challenge outcome measures</b> (A) To measure oxazolone-driven vulvar tactile sensitivity, mice were topically sensitized with 2% Ox on the shaved flank (day 1) and subsequently challenged on the shaved labiar skin (days 5–14) with 1% Ox or EtOH vehicle for a total of 10 challenges. Tactile sensitivity was assessed in the ano-genital ridge area 1, 21, and 42 days after challenge cessation. Labiar skin was harvested at these time points from a separate cohort of mice for assessing molecular and cellular changes in the tissue. (B) To characterize T cell infiltration in Ox-challenged skin, mice were topically sensitized on their shaved back with 2% Ox (day 1) and challenged on both shaved flanks with 1% Ox or EtOH (days 5–14). Flank skin was harvested from both sides 1 day after challenge cessation for flow cytometric analysis of T cell infiltration. (C) To assess the effects of local mast cell depletion on Ox-induced tactile sensitivity and hyperinnervation, mice were topically sensitized with 2% Ox on the shaved flank (day 1), challenged on the shaved labia with 1% Ox or EtOH (days 5–14) and treated with intralabiar injection of saline or c48/80 (days 5–8 after Ox challenge cessation). Tactile sensitivity, mast cell levels and innervation were assessed 9, 21, and 35 days after the final Ox challenge.</p

Injection of c48/80 after challenges depletes mast cells and reduces CGRP⁺ nerve density and sensitivity.

<p>Density of Avidin⁺ mast cells (A) and CGRP⁺ cutaneous nerves (D) on day 9 after 4 treatments with c48/80 or saline (administered on days 5–8 after cessation of 10 Ox challenges) in 10 μm labiar cryo-sections, displayed as mean ± SEM (n = 2-3/treatment group). Representative images for mast cells (B-C) and nerves (E-F); 20x magnification; scale bar represents 50 μm. Means are compared to Ox/EtOH (** = p<0.01, *** = p<0.001); significance determined using one-way ANOVA and Tukey Kramer <i>post hoc</i> analysis. (G) Tactile sensitivity in mice treated with either saline or c48/80 (n = 6–9 mice per treatment group; two independent experiments). Means are compared to Ox/Ox/Saline (* = p<0.05, ** = p<0.01, *** = p<0.001); significance determined using an unpaired Student’s T test at each time point.</p

Injection of c48/80 after challenges depletes mast cells and reduces CGRP⁺ nerve density and sensitivity.

<p>Density of Avidin⁺ mast cells (A) and CGRP⁺ cutaneous nerves (D) on day 9 after 4 treatments with c48/80 or saline (administered on days 5–8 after cessation of 10 Ox challenges) in 10 μm labiar cryo-sections, displayed as mean ± SEM (n = 2-3/treatment group). Representative images for mast cells (B-C) and nerves (E-F); 20x magnification; scale bar represents 50 μm. Means are compared to Ox/EtOH (** = p<0.01, *** = p<0.001); significance determined using one-way ANOVA and Tukey Kramer <i>post hoc</i> analysis. (G) Tactile sensitivity in mice treated with either saline or c48/80 (n = 6–9 mice per treatment group; two independent experiments). Means are compared to Ox/Ox/Saline (* = p<0.05, ** = p<0.01, *** = p<0.001); significance determined using an unpaired Student’s T test at each time point.</p

Ten oxazolone challenges provoke tactile sensitivity that persists for 21 days after cessation of challenges.

<p>Sensitized mice that received ten daily Ox challenges on the labiar skin had increased tactile sensitivity in their ano-genital ridge area compared to controls. Percent decrease in labiar withdrawal threshold for each treatment group is displayed as mean ± SEM. Significance at each time point was determined using one-way ANOVA and Tukey Kramer <i>post hoc</i> analysis based on comparisons to previously sensitized mice challenged with EtOH (Ox/EtOH; * = p<0.05, *** = p<0.001) and untreated controls (NT; ## = p<0.01, ### = p<0.001). n = 9–12 mice per treatment group; data represent two independent experiments. Raw withdrawal thresholds at baseline and post Ox-challenge cessation for each animal are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0169672#pone.0169672.s001" target="_blank">S1 Fig</a> and summarized in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0169672#pone.0169672.s004" target="_blank">S1 Table</a>.</p

CD4⁺CD25⁺FoxP3⁺ T cells and IFN-γ producing CD8⁺CD103⁺ memory T cells accumulate in oxazolone challenged skin.

<p>Flow cytometric analysis of collagenase-digested, gradient-separated flank skin cells from Ox- (A-B) or EtOH- (C-D) challenged mice 1 day after the cessation of 10 challenges. Cells in (A) and (C) are scatter-gated for lymphocytes, single cells, live, and CD45⁺. Data are pooled from 10 mice per treatment group. Relative abundance of <i>Ifn-γ</i> and <i>Tbx21</i> (E) and total IFN-<i>γ</i> protein content (F) in labiar skin of Ox- vs. EtOH challenged mice after 10 challenges displayed as mean ± SEM (n = 5-6/treatment group). (G-H) Flow cytometric analysis of collagenase-digested, gradient-separated flank skin cells collected 1 day after 10 challenges from Ox-challenged mice and cultured for 18 hours with or without PMA/ionomycin; cells in (G) are live, singlet-gated, scatter-gated for lymphocytes, and CD45⁺CD3⁺CD8⁺.</p