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

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

A genome-wide association study of anorexia nervosa.

Anorexia nervosa (AN) is a complex and heritable eating disorder characterized by dangerously low body weight. Neither candidate gene studies nor an initial genome-wide association study (GWAS) have yielded significant and replicated results. We performed a GWAS in 2907 cases with AN from 14 countries (15 sites) and 14 860 ancestrally matched controls as part of the Genetic Consortium for AN (GCAN) and the Wellcome Trust Case Control Consortium 3 (WTCCC3). Individual association analyses were conducted in each stratum and meta-analyzed across all 15 discovery data sets. Seventy-six (72 independent) single nucleotide polymorphisms were taken forward for in silico (two data sets) or de novo (13 data sets) replication genotyping in 2677 independent AN cases and 8629 European ancestry controls along with 458 AN cases and 421 controls from Japan. The final global meta-analysis across discovery and replication data sets comprised 5551 AN cases and 21 080 controls. AN subtype analyses (1606 AN restricting; 1445 AN binge-purge) were performed. No findings reached genome-wide significance. Two intronic variants were suggestively associated: rs9839776 (P=3.01 × 10(-7)) in SOX2OT and rs17030795 (P=5.84 × 10(-6)) in PPP3CA. Two additional signals were specific to Europeans: rs1523921 (P=5.76 × 10(-)(6)) between CUL3 and FAM124B and rs1886797 (P=8.05 × 10(-)(6)) near SPATA13. Comparing discovery with replication results, 76% of the effects were in the same direction, an observation highly unlikely to be due to chance (P=4 × 10(-6)), strongly suggesting that true findings exist but our sample, the largest yet reported, was underpowered for their detection. The accrual of large genotyped AN case-control samples should be an immediate priority for the field

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

A genome-wide association study of anorexia nervosa

Anorexia nervosa (AN) is a complex and heritable eating disorder characterized by dangerously low body weight. Neither candidate gene studies nor an initial genome wide association study (GWAS) have yielded significant and replicated results. We performed a GWAS in 2,907 cases with AN from 14 countries (15 sites) and 14,860 ancestrally matched controls as part of the Genetic Consortium for AN (GCAN) and the Wellcome Trust Case Control Consortium 3 (WTCCC3). Individual association analyses were conducted in each stratum and meta-analyzed across all 15 discovery datasets. Seventy-six (72 independent) SNPs were taken forward for in silico (two datasets) or de novo (13 datasets) replication genotyping in 2,677 independent AN cases and 8,629 European ancestry controls along with 458 AN cases and 421 controls from Japan. The final global meta-analysis across discovery and replication datasets comprised 5,551 AN cases and 21,080 controls. AN subtype analyses (1,606 AN restricting; 1,445 AN binge-purge) were performed. No findings reached genome-wide significance. Two intronic variants were suggestively associated: rs9839776 (P=3.01×10−7) in SOX2OT and rs17030795 (P=5.84×10−6) in PPP3CA. Two additional signals were specific to Europeans: rs1523921 (P=5.76×10−6) between CUL3 and FAM124B and rs1886797 (P=8.05×10−6) near SPATA13. Comparing discovery to replication results, 76% of the effects were in the same direction, an observation highly unlikely to be due to chance (P= 4×10−6), strongly suggesting that true findings exist but that our sample, the largest yet reported, was underpowered for their detection. The accrual of large genotyped AN case-control samples should be an immediate priority for the field

Is there a protective effect of normal to high intellectual function on mental health in children with chronic illness?

<p>Abstract</p> <p>Background</p> <p>High intellectual function is considered as a protective factor for children's mental health. Few studies have investigated the effect of intellectual function on mental health in children with chronic illness (CI). The aim of the present study was twofold: First, we asked if <it>normal to high </it>intellectual function (IQ) has a protective effect on mental health in children with CI, and secondly, if this effect is more substantial than in their peers (NCI).</p> <p>Methods</p> <p>The participants were selected among children who participated in the Bergen Child Study (BCS): 96 children with CI (the CI-group) and 96 children without CI (the NCI-group). The groups were matched on intellectual function as measured by the WISC-III by selecting the same number of children from three levels of the Full Scale IQ Score (FSIQ): "very low" (<70),"low" (70 to 84), or "normal to high" (>84). CI was reported by parents as part of a diagnostic interview (Kiddie-SADS-PL) that also generated the mental health measures used in the present study: the presence of a DSM-IV psychiatric diagnosis and the score on the Children's Global Assessment Scale.</p> <p>Results</p> <p>The risk of a psychiatric diagnosis was significantly lower for children with a normal to high FSIQ-level than for children with a very low and low FSIQ-level in the CI-group as well as in the NCI-group. The group differences were statistically non-significant for all three FSIQ-levels, and the effect of the interaction between the group-variable (CI/NCI) and the FSIQ-level was non-significant on both measures of mental health.</p> <p>Conclusion</p> <p>The present study showed a protective effect of normal to high intellectual function on children's mental health. This protective effect was not more substantial in children with CI than in children without CI.</p

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

Dynamics of affect modulation in neurodevelopmental disorders (DynAMoND) – study design of a prospective cohort study

Background Attention-deficit/hyperactivity disorder (ADHD) is a common neuro-developmental disorder that often persists into adulthood. Moreover, it is frequently accompanied by bipolar disorder (BD) as well as borderline personality disorder (BPD). It is unclear whether these disorders share underlying pathomechanisms, given that all three are characterized by alterations in affective states, either long or short-term. BD is characterized by infrequent but intense mood shifts, while ADHD and BPD involve more dynamic emotional fluctuations. It is yet to be determined whether these disorders represent distinct phenomena or different points on a spectrum of affective dysregulation. Methods This study seeks to distinguish the emotional dysregulation of BPD, ADHD, and BD by using digital phenotyping, a measurement burst electronic-diary method with different sampling rates, and accelerometry to measure participants’ activity. Our study will include 480 participants aged 14 to 50 (120 each from BPD, ADHD, BD, and healthy control groups) from five European sites. Participants’ smartphones will provide continuous data on their digital phenotypes, i.e., by indicators of physical activity and communication, for one year, along with daily evening ratings of mood and sleep. Moreover, five intensive measurement periods of five days each, called measurement bursts, will occur throughout the year, with electronic diaries asking participants to report on mood, self-esteem, impulsivity, life events, social interactions, and dysfunctional behaviors ten times a day. Moreover, participants will wear activity sensors during the five measurement bursts. Statistical analysis aims to identify whether affective dysregulation aspects share or differ across disorders. Specifically, data analysis aims to investigate the differences in parameters of affect fluctuation such as attractor strength and variability between disorders and to test the association of genetic risk factors for psychiatric disorders and resilience factors with critical parameters of affect modulation. Discussion The results of this study offer the potential to link patients’ external exposures with their affective state, reduce misdiagnosis, and determine the best timing for therapeutic interventions. Potential limitations of the study include insufficient recruitment of patients and drop-outs due to various protocol violations