Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

On post-Lie algebras, Lie--Butcher series and moving frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie--Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie--Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie--Butcher series are related to invariants of curves described by moving frames.Comment: added discussion of post-Lie algebroid

Teacher reports of hypoactivity symptoms reflect slow cognitive processing speed in primary school children

The mediating effect of cognitive processing speed on the ability of a primary school child to achieve his/her full potential of intellectual functioning emphasizes the importance of methods to detect “slow” children. Primary school teachers may be the first to have concerns about inattentive pupils who show symptoms of hypoactivity, but may find the symptoms difficult to interpret. In the present study we ask if a primary school teacher’s report of hypoactivity symptoms can be explained by the child’s performance on tests of processing speed. The 255 children included in the present study were part of the first wave of the Bergen Child Study, in which teachers completed a questionnaire including two hypoactivity items from the Five to Fifteen (FTF) questionnaire. Processing speed was measured by the Processing Speed Index (PSI) from the WISC-III, 1–2 years after the teacher rating. Teachers reported “certainly true” on at least one FTF item of hypoactivity for 11.8% of the children. These children obtained lower scores on the PSI than the remaining children in the sample. The PSI accounted for a considerable proportion of the variance of teacher reports on the FTF item “difficulty getting started on a task/activity”. The risk of a PSI score below 85 was increased in children with teacher-reported hypoactivity symptoms. The results indicate that teacher reports of hypoactivity symptoms reflect slow cognitive processing speed and should be followed up by a psychometric examination. Still, future studies are needed to improve detection and treatment of children with slow processing speed

A Genetic Deconstruction of Neurocognitive Traits in Schizophrenia and Bipolar Disorder

Background: Impairments in cognitive functions are common in patients suffering from psychiatric disorders, such as schizophrenia and bipolar disorder. Cognitive traits have been proposed as useful for understanding the biological and genetic mechanisms implicated in cognitive function in healthy individuals and in the dysfunction observed in psychiatric disorders. Methods: Sets of genes associated with a range of cognitive functions often impaired in schizophrenia and bipolar disorder were generated from a genome-wide association study (GWAS) on a sample comprising 670 healthy Norwegian adults who were phenotyped for a broad battery of cognitive tests. These gene sets were then tested for enrichment of association in GWASs of schizophrenia and bipolar disorder. The GWAS data was derived from three independent single-centre schizophrenia samples, three independent single-centre bipolar disorder samples, and the multi-centre schizophrenia and bipolar disorder samples from the Psychiatric Genomics Consortium. Results: The strongest enrichments were observed for visuospatial attention and verbal abilities sets in bipolar disorder. Delayed verbal memory was also enriched in one sample of bipolar disorder. For schizophrenia, the strongest evidence of enrichment was observed for the sets of genes associated with performance in a colour-word interference test and for sets associated with memory learning slope. Conclusions: Our results are consistent with the increasing evidence that cognitive functions share genetic factors with schizophrenia and bipolar disorder. Our data provides evidence that genetic studies using polygenic and pleiotropic models can be used to link specific cognitive functions with psychiatric disorders

Hippocampal volumes are important predictors for memory function in elderly women

<p>Abstract</p> <p>Background</p> <p>Normal aging involves a decline in cognitive function that has been shown to correlate with volumetric change in the hippocampus, and with genetic variability in the APOE-gene. In the present study we utilize 3D MR imaging, genetic analysis and assessment of verbal memory function to investigate relationships between these factors in a sample of 170 healthy volunteers (age range 46–77 years).</p> <p>Methods</p> <p>Brain morphometric analysis was performed with the automated segmentation work-flow implemented in FreeSurfer. Genetic analysis of the APOE genotype was determined with polymerase chain reaction (PCR) on DNA from whole-blood. All individuals were subjected to extensive neuropsychological testing, including the California Verbal Learning Test-II (CVLT). To obtain robust and easily interpretable relationships between explanatory variables and verbal memory function we applied the recent method of conditional inference trees in addition to scatterplot matrices and simple pairwise linear least-squares regression analysis.</p> <p>Results</p> <p>APOE genotype had no significant impact on the CVLT results (scores on long delay free recall, CVLT-LD) or the ICV-normalized hippocampal volumes. Hippocampal volumes were found to decrease with age and a right-larger-than-left hippocampal asymmetry was also found. These findings are in accordance with previous studies. CVLT-LD score was shown to correlate with hippocampal volume. Multivariate conditional inference analysis showed that gender and left hippocampal volume largely dominated predictive values for CVLT-LD scores in our sample. Left hippocampal volume dominated predictive values for females but not for males. APOE genotype did not alter the model significantly, and age was only partly influencing the results.</p> <p>Conclusion</p> <p>Gender and left hippocampal volumes are main predictors for verbal memory function in normal aging. APOE genotype did not affect the results in any part of our analysis.</p

Effects of copy number variations on brain structure and risk for psychiatric illness: Large-scale studies from the ENIGMA working groups on CNVs

The Enhancing NeuroImaging Genetics through Meta-Analysis copy number variant (ENIGMA-CNV) and 22q11.2 Deletion Syndrome Working Groups (22q-ENIGMA WGs) were created to gain insight into the involvement of genetic factors in human brain development and related cognitive, psychiatric and behavioral manifestations. To that end, the ENIGMA-CNV WG has collated CNV and magnetic resonance imaging (MRI) data from ~49,000 individuals across 38 global research sites, yielding one of the largest studies to date on the effects of CNVs on brain structures in the general population. The 22q-ENIGMA WG includes 12 international research centers that assessed over 533 individuals with a confirmed 22q11.2 deletion syndrome, 40 with 22q11.2 duplications, and 333 typically developing controls, creating the largest-ever 22q11.2 CNV neuroimaging data set. In this review, we outline the ENIGMA infrastructure and procedures for multi-site analysis of CNVs and MRI data. So far, ENIGMA has identified effects of the 22q11.2, 16p11.2 distal, 15q11.2, and 1q21.1 distal CNVs on subcortical and cortical brain structures. Each CNV is associated with differences in cognitive, neurodevelopmental and neuropsychiatric traits, with characteristic patterns of brain structural abnormalities. Evidence of gene-dosage effects on distinct brain regions also emerged, providing further insight into genotype–phenotype relationships. Taken together, these results offer a more comprehensive picture of molecular mechanisms involved in typical and atypical brain development. This “genotype-first” approach also contributes to our understanding of the etiopathogenesis of brain disorders. Finally, we outline future directions to better understand effects of CNVs on brain structure and behavior

GWAS meta-analysis reveals novel loci and genetic correlates for general cognitive function : a report from the COGENT consortium

CORRIGENDUM Molecular Psychiatry (2017) 22, 1651–1652 http://www.nature.com/articles/mp2017197.pdfThe complex nature of human cognition has resulted in cognitive genomics lagging behind many other fields in terms of gene discovery using genome-wide association study (GWAS) methods. In an attempt to overcome these barriers, the current study utilized GWAS meta-analysis to examine the association of common genetic variation (similar to 8M single-nucleotide polymorphisms (SNP) with minor allele frequency >= 1%) to general cognitive function in a sample of 35 298 healthy individuals of European ancestry across 24 cohorts in the Cognitive Genomics Consortium (COGENT). In addition, we utilized individual SNP lookups and polygenic score analyses to identify genetic overlap with other relevant neurobehavioral phenotypes. Our primary GWAS meta-analysis identified two novel SNP loci (top SNPs: rs76114856 in the CENPO gene on chromosome 2 and rs6669072 near LOC105378853 on chromosome 1) associated with cognitive performance at the genome-wide significance level (PPeer reviewe