Renormalization Group Theory for Global Asymptotic Analysis

We show with several examples that renormalization group (RG) theory can be used to understand singular and reductive perturbation methods in a unified fashion. Amplitude equations describing slow motion dynamics in nonequilibrium phenomena are RG equations. The renormalized perturbation approach may be simpler to use than other approaches, because it does not require the use of asymptotic matching, and yields practically superior approximations.Comment: 13 pages, plain tex + uiucmac.tex (available from babbage.sissa.it), one PostScript figure appended at end. Or (easier) get compressed postscript file by anon ftp from gijoe.mrl.uiuc.edu (128.174.119.153), file /pub/rg_sing_prl.ps.

Kinetics of Ordering in Fluctuation-Driven First-Order Transitions: Simulations and Dynamical Renormalization

Many systems where interactions compete with each other or with constraints are well described by a model first introduced by Brazovskii. Such systems include block copolymers, alloys with modulated phases, Rayleigh-Benard Cells and type-I superconductors. The hallmark of this model is that the fluctuation spectrum is isotropic and has a minimum at a nonzero wave vector represented by the surface of a d-dimensional hyper-sphere. It was shown by Brazovskii that the fluctuations change the free energy structure from a $\phi ^{4}$ to a $\phi ^{6}$ form with the disordered state metastable for all quench depths. The transition from the disordered to the periodic, lamellar structure changes from second order to first order and suggests that the dynamics is governed by nucleation. Using numerical simulations we have confirmed that the equilibrium free energy function is indeed of a $\phi ^{6}$ form. A study of the dynamics, however, shows that, following a deep quench, the dynamics is described by unstable growth rather than nucleation. A dynamical calculation, based on a generalization of the Brazovskii calculations shows that the disordered state can remain unstable for a long time following the quench.Comment: 18 pages, 15 figures submitted to PR

The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither {\it ad hoc\/} assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially-extended systems near bifurcation points, deriving both amplitude equations and the center manifold.Comment: 44 pages, 2 Postscript figures, macro \uiucmac.tex available at macro archives or at ftp://gijoe.mrl.uiuc.edu/pu

Coordinated optimization of visual cortical maps (I) Symmetry-based analysis

In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of OP columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about an hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference.Comment: 90 pages, 16 figure

Coordinated optimization of visual cortical maps (II) Numerical studies

It is an attractive hypothesis that the spatial structure of visual cortical architecture can be explained by the coordinated optimization of multiple visual cortical maps representing orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference. In part (I) of this study we defined a class of analytically tractable coordinated optimization models and solved representative examples in which a spatially complex organization of the orientation preference map is induced by inter-map interactions. We found that attractor solutions near symmetry breaking threshold predict a highly ordered map layout and require a substantial OD bias for OP pinwheel stabilization. Here we examine in numerical simulations whether such models exhibit biologically more realistic spatially irregular solutions at a finite distance from threshold and when transients towards attractor states are considered. We also examine whether model behavior qualitatively changes when the spatial periodicities of the two maps are detuned and when considering more than 2 feature dimensions. Our numerical results support the view that neither minimal energy states nor intermediate transient states of our coordinated optimization models successfully explain the spatially irregular architecture of the visual cortex. We discuss several alternative scenarios and additional factors that may improve the agreement between model solutions and biological observations.Comment: 55 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:1102.335

Genetic risk and a primary role for cell-mediated immune mechanisms in multiple sclerosis.

Multiple sclerosis is a common disease of the central nervous system in which the interplay between inflammatory and neurodegenerative processes typically results in intermittent neurological disturbance followed by progressive accumulation of disability. Epidemiological studies have shown that genetic factors are primarily responsible for the substantially increased frequency of the disease seen in the relatives of affected individuals, and systematic attempts to identify linkage in multiplex families have confirmed that variation within the major histocompatibility complex (MHC) exerts the greatest individual effect on risk. Modestly powered genome-wide association studies (GWAS) have enabled more than 20 additional risk loci to be identified and have shown that multiple variants exerting modest individual effects have a key role in disease susceptibility. Most of the genetic architecture underlying susceptibility to the disease remains to be defined and is anticipated to require the analysis of sample sizes that are beyond the numbers currently available to individual research groups. In a collaborative GWAS involving 9,772 cases of European descent collected by 23 research groups working in 15 different countries, we have replicated almost all of the previously suggested associations and identified at least a further 29 novel susceptibility loci. Within the MHC we have refined the identity of the HLA-DRB1 risk alleles and confirmed that variation in the HLA-A gene underlies the independent protective effect attributable to the class I region. Immunologically relevant genes are significantly overrepresented among those mapping close to the identified loci and particularly implicate T-helper-cell differentiation in the pathogenesis of multiple sclerosis

Multi-trait analysis characterizes the genetics of thyroid function and identifies causal associations with clinical implications

To date only a fraction of the genetic footprint of thyroid function has been clarified. We report a genome-wide association study meta-analysis of thyroid function in up to 271,040 individuals of European ancestry, including reference range thyrotropin (TSH), free thyroxine (FT4), free and total triiodothyronine (T3), proxies for metabolism (T3/FT4 ratio) as well as dichotomized high and low TSH levels. We revealed 259 independent significant associations for TSH (61% novel), 85 for FT4 (67% novel), and 62 novel signals for the T3 related traits. The loci explained 14.1%, 6.0%, 9.5% and 1.1% of the total variation in TSH, FT4, total T3 and free T3 concentrations, respectively. Genetic correlations indicate that TSH associated loci reflect the thyroid function determined by free T3, whereas the FT4 associations represent the thyroid hormone metabolism. Polygenic risk score and Mendelian randomization analyses showed the effects of genetically determined variation in thyroid function on various clinical outcomes, including cardiovascular risk factors and diseases, autoimmune diseases, and cancer. In conclusion, our results improve the understanding of thyroid hormone physiology and highlight the pleiotropic effects of thyroid function on various diseases.</p

Multi-trait analysis characterizes the genetics of thyroid function and identifies causal associations with clinical implications

To date only a fraction of the genetic footprint of thyroid function has been clarified. We report a genome-wide association study meta-analysis of thyroid function in up to 271,040 individuals of European ancestry, including reference range thyrotropin (TSH), free thyroxine (FT4), free and total triiodothyronine (T3), proxies for metabolism (T3/FT4 ratio) as well as dichotomized high and low TSH levels. We revealed 259 independent significant associations for TSH (61% novel), 85 for FT4 (67% novel), and 62 novel signals for the T3 related traits. The loci explained 14.1%, 6.0%, 9.5% and 1.1% of the total variation in TSH, FT4, total T3 and free T3 concentrations, respectively. Genetic correlations indicate that TSH associated loci reflect the thyroid function determined by free T3, whereas the FT4 associations represent the thyroid hormone metabolism. Polygenic risk score and Mendelian randomization analyses showed the effects of genetically determined variation in thyroid function on various clinical outcomes, including cardiovascular risk factors and diseases, autoimmune diseases, and cancer. In conclusion, our results improve the understanding of thyroid hormone physiology and highlight the pleiotropic effects of thyroid function on various diseases.</p

Incorporating progesterone receptor expression into the PREDICT breast prognostic model

Background: Predict Breast (www.predict.nhs.uk) is an online prognostication and treatment benefit tool for early invasive breast cancer. The aim of this study was to incorporate the prognostic effect of progesterone receptor (PR) status into a new version of PREDICT and to compare its performance to the current version (2.2).Method: The prognostic effect of PR status was based on the analysis of data from 45,088 European patients with breast cancer from 49 studies in the Breast Cancer Association Consortium. Cox proportional hazard models were used to estimate the hazard ratio for PR status. Data from a New Zealand study of 11,365 patients with early invasive breast cancer were used for external validation. Model calibration and discrimination were used to test the model performance.Results: Having a PR-positive tumour was associated with a 23% and 28% lower risk of dying from breast cancer for women with oestrogen receptor (ER)-negative and ER-positive breast cancer, respectively. The area under the ROC curve increased with the addition of PR status from 0.807 to 0.809 for patients with ER-negative tumours (p = 0.023) and from 0.898 to 0. 902 for patients with ER-positive tumours (p = 2.3 x 10(-6)) in the New Zealand cohort. Model calibration was modest with 940 observed deaths compared to 1151 predicted.Conclusion: The inclusion of the prognostic effect of PR status to PREDICT Breast has led to an improvement of model performance and more accurate absolute treatment benefit predic-tions for individual patients. Further studies should determine whether the baseline hazard function requires recalibration. (C) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Peer reviewe