Selected Publications

Lee, P., et al. Genomic relationships, novel loci, and pleiotropic mechanisms across eight psychiatric disorders. Cell 179, 1469-1482 (2019).Abstract

Genetic influences on psychiatric disorders transcend diagnostic boundaries, suggesting substantial pleiotropy of contributing loci. However, the nature and mechanisms of these pleiotropic effects remain unclear. We performed analyses of 232,964 cases and 494,162 controls from genome-wide studies of anorexia nervosa, attention-deficit/hyper-activity disorder, autism spectrum disorder, bipolar disorder, major depression, obsessive-compulsive disorder, schizophrenia, and Tourette syndrome. Genetic correlation analyses revealed a meaningful structure within the eight disorders, identifying three groups of inter-related disorders. Meta-analysis across these eight disorders detected 109 loci associated with at least two psychiatric disorders, including 23 loci with pleiotropic effects on four or more disorders and 11 loci with antagonistic effects on multiple disorders. The pleiotropic loci are located within genes that show heightened expression in the brain throughout the lifespan, beginning prenatally in the second trimester, and play prominent roles in neurodevelopmental processes. These findings have important implications for psychiatric nosology, drug development, and risk prediction.

Zheutlin, A., et al. Penetrance and Pleiotropy of Polygenic Risk Scores for Schizophrenia in 106,160 Patients Across Four Health Care Systems. American Journal of Psychiatry 176, 10, 846-855 (2019).Abstract

Objective: Individuals at high risk for schizophrenia may benefit from early intervention, but few validated risk predictors are available. Genetic profiling is one approach to risk stratification that has been extensively validated in research cohorts. The authors sought to test the utility of this approach in clinical settings and to evaluate the broader health consequences of high genetic risk for schizophrenia.

Methods: The authors used electronic health records for 106,160 patients from four health care systems to evaluate the penetrance and pleiotropy of genetic risk for schizophrenia. Polygenic risk scores (PRSs) for schizophrenia were calculated from summary statistics and tested for association with 1,359 disease categories, including schizophrenia and psychosis, in phenome-wide association studies. Effects were combined through meta-analysis across sites.

Results: PRSs were robustly associated with schizophrenia (odds ratio per standard deviation increase in PRS, 1.55; 95% CI=1.4, 1.7), and patients in the highest risk decile of the PRS distribution had up to 4.6-fold higher odds of schizophrenia compared with those in the bottom decile (95% CI=2.9, 7.3). PRSs were also positively associated with other phenotypes, including anxiety, mood, substance use, neurological, and personality disorders, as well as suicidal behavior, memory loss, and urinary syndromes; they were inversely related to obesity.

Conclusions: The study demonstrates that an available measure of genetic risk for schizophrenia is robustly associated with schizophrenia in health care settings and has pleiotropic effects on related psychiatric disorders as well as other medical syndromes. The results provide an initial indication of the opportunities and limitations that may arise with the future application of PRS testing in health care systems.

Liégeois, R., et al. Resting brain dynamics at different timescales capture distinct aspects of human behavior. Nature Communications 10, 2317 (2019).Abstract

Linking human behavior to resting-state brain function is a central question in systems neuroscience. In particular, the functional timescales at which different types of behavioral factors are encoded remain largely unexplored. The behavioral counterparts of static functional connectivity (FC), at the resolution of several minutes, have been studied but behavioral correlates of dynamic measures of FC at the resolution of a few seconds remain unclear. Here, using resting-state fMRI and 58 phenotypic measures from the Human Connectome Project, we find that dynamic FC captures task-based phenotypes (e.g., processing speed or fluid intelligence scores), whereas self-reported measures (e.g., loneliness or life satisfaction) are equally well explained by static and dynamic FC. Furthermore, behaviorally relevant dynamic FC emerges from the interconnections across all resting-state networks, rather than within or between pairs of networks. Our findings shed new light on the timescales of cognitive processes involved in distinct facets of behavior.

Li, J., et al. Global signal regression strengthens association between resting-state functional connectivity and behavior. NeuroImage 196, 126-141 (2019).Abstract

Global signal regression (GSR) is one of the most debated preprocessing strategies for resting-state functional MRI. GSR effectively removes global artifacts driven by motion and respiration, but also discards globally distributed neural information and introduces negative correlations between certain brain regions. The vast majority of previous studies have focused on the effectiveness of GSR in removing imaging artifacts, as well as its potential biases. Given the growing interest in functional connectivity fingerprinting, here we considered the utilitarian question of whether GSR strengthens or weakens associations between resting-state functional connectivity (RSFC) and multiple behavioral measures across cognition, personality and emotion.

By applying the variance component model to the Brain Genomics Superstruct Project (GSP), we found that behavioral variance explained by whole-brain RSFC increased by an average of 47% across 23 behavioral measures after GSR. In the Human Connectome Project (HCP), we found that behavioral variance explained by whole-brain RSFC increased by an average of 40% across 58 behavioral measures, when GSR was applied after ICA-FIX de-noising. To ensure generalizability, we repeated our analyses using kernel regression. GSR improved behavioral prediction accuracies by an average of 64% and 12% in the GSP and HCP datasets respectively. Importantly, the results were consistent across methods. A behavioral measure with greater RSFC-explained variance (using the variance component model) also exhibited greater prediction accuracy (using kernel regression). A behavioral measure with greater improvement in behavioral variance explained after GSR (using the variance component model) also enjoyed greater improvement in prediction accuracy after GSR (using kernel regression). Furthermore, GSR appeared to benefit task performance measures more than self-reported measures.

Since GSR was more effective at removing motion-related and respiratory-related artifacts, GSR-related increases in variance explained and prediction accuracies were unlikely the result of motion-related or respiratory- related artifacts. However, it is worth emphasizing that the current study focused on whole-brain RSFC, so it remains unclear whether GSR improves RSFC-behavioral associations for specific connections or networks. Overall, our results suggest that at least in the case for young healthy adults, GSR strengthens the associations between RSFC and most (although not all) behavioral measures. Code for the variance component model and ridge regression can be found here:

Ge, T., Chen, C.-Y., Ni, Y., Feng, Y.-C.A. & Smoller, J.W. Polygenic prediction via Bayesian regression and continuous shrinkage priors. Nature Communications 10, 1776 (2019).Abstract
Polygenic risk scores (PRS) have shown promise in predicting human complex traits and diseases. Here, we present PRS-CS, a polygenic prediction method that infers posterior effect sizes of single nucleotide polymorphisms (SNPs) using genome-wide association summary statistics and an external linkage disequilibrium (LD) reference panel. PRS-CS utilizes a high-dimensional Bayesian regression framework, and is distinct from previous work by placing a continuous shrinkage (CS) prior on SNP effect sizes, which is robust to varying genetic architectures, provides substantial computational advantages, and enables multivariate modeling of local LD patterns. Simulation studies using data from the UK Biobank show that PRS-CS outperforms existing methods across a wide range of genetic architectures, especially when the training sample size is large. We apply PRS-CS to predict six common complex diseases and six quantitative traits in the Partners HealthCare Biobank, and further demonstrate the improvement of PRS-CS in prediction accuracy over alternative methods.
Ge, T., et al. The Shared Genetic Basis of Educational Attainment and Cerebral Cortical Morphology. Cerebral Cortex 29, 8, 3471-3481 (2018).Abstract

Individual differences in educational attainment are linked to differences in intelligence, and predict important social, economic, and health outcomes. Previous studies have found common genetic factors that influence educational achievement, cognitive performance and total brain volume (i.e., brain size). Here, in a large sample of participants from the UK Biobank, we investigate the shared genetic basis between educational attainment and fine-grained cerebral cortical morphological features, and associate this genetic variation with a related aspect of cognitive ability. Importantly, we execute novel statistical methods that enable high-dimensional genetic correlation analysis, and compute high-resolution surface maps for the genetic correlations between educational attainment and vertex-wise morphological measurements. We conduct secondary analyses, using the UK Biobank verbal–numerical reasoning score, to confirm that variation in educational attainment that is genetically correlated with cortical morphology is related to differences in cognitive performance. Our analyses relate the genetic overlap between cognitive ability and cortical thickness measurements to bilateral primary motor cortex as well as predominantly left superior temporal cortex and proximal regions. These findings extend our understanding of the neurobiology that connects genetic variation to individual differences in educational attainment and cognitive performance.

Ge, T., Subuncu, M.R., Smoller, J.W., Sperling, R.A. & Mormino, E.C. Dissociable influences of APOE ε4 and polygenic risk of AD dementia on amyloid and cognition. Neurology 90, 18, e1605-e1612 (2018).Abstract

Objective -- To investigate the effects of genetic risk of Alzheimer disease (AD) dementia in the context of β-amyloid (Aβ) accumulation.

Methods -- We analyzed data from 702 participants (221 clinically normal, 367 with mild cognitive impairment, and 114 with AD dementia) with genetic data and florbetapir PET available. A subset of 669 participants additionally had longitudinal MRI scans to assess hippocampal volume. Polygenic risk scores (PRSs) were estimated with summary statistics from previous large-scale genome-wide association studies of AD dementia. We examined relationships between APOEe4 status and PRS with longitudinal Aβ and cognitive and hippocampal volume measurements.

Results -- APOE e4 was strongly related to baseline Aβ, whereas only weak associations between PRS and baseline Aβ were present. APOE e4 was additionally related to greater memory decline and hippocampal atrophy in Aβ+ participants. When APOE e4 was controlled for, PRS was related to cognitive decline in Aβ+ participants. Finally, PRSs were associated with hippocampal atrophy in Aβ− participants and weakly associated with baseline hippocampal volume in Aβ+ participants.

Conclusions -- Genetic risk factors of AD dementia demonstrate effects related to Aβ, as well as synergistic interactions with Aβ. The specific effect of faster cognitive decline in Aβ+ individuals with higher genetic risk may explain the large degree of heterogeneity in cognitive trajectories among Aβ+ individuals. Consideration of genetic variants in conjunction with baseline Aβ may improve enrichment strategies for clinical trials targeting Aβ+ individuals most at risk for imminent cognitive decline.

Ge, T., Holmes, A.J., Buckner, R.L., Smoller, J.W. & Sabuncu, M.R. Heritability analysis with repeat measurements and its application to resting-state functional connectivity. Proceedings of the National Academy of Sciences 114, 21, 5521–5526 (2017).Abstract

Heritability, defined as the proportion of phenotypic variation attributable to genetic variation, provides important information about the genetic basis of a trait. Existing heritability analysis methods do not discriminate between stable effects (e.g., due to the subject’s unique environment) and transient effects, such as measurement error. This can lead to misleading assessments, particularly when comparing the heritability of traits that exhibit different levels of reliability. Here, we present a linear mixed effects model to conduct heritability analyses that explicitly accounts for intrasubject fluctuations (e.g., due to measurement noise or biological transients) using repeat measurements. We apply the proposed strategy to the analysis of resting-state fMRI measurements—a prototypic data modality that exhibits variable levels of test–retest reliability across space. Our results reveal that the stable components of functional connectivity within and across well-established large-scale brain networks can be considerably heritable. Furthermore, we demonstrate that dissociating intra- and intersubject variation can reveal genetic influence on a phenotype that is not fully captured by conventional heritability analyses.

Ge, T., Chen, C.-Y., Neale, B.M., Sabuncu, M.R. & Smoller, J.W. Phenome-wide Heritability Analysis of the UK Biobank. PLOS Genetics 13, 4, e1006711 (2017).Abstract

Heritability estimation provides important information about the relative contribution of genetic and environmental factors to phenotypic variation, and provides an upper bound for the utility of genetic risk prediction models. Recent technological and statistical advances have enabled the estimation of additive heritability attributable to common genetic variants (SNP heritability) across a broad phenotypic spectrum. Here, we present a computationally and memory efficient heritability estimation method that can handle large sample sizes, and report the SNP heritability for 551 complex traits derived from the interim data release (152,736 subjects) of the large-scale, population-based UK Biobank, comprising both quantitative phenotypes and disease codes. We demonstrate that common genetic variation contributes to a broad array of quantitative traits and human diseases in the UK population, and identify phenotypes whose heritability is moderated by age (e.g., a majority of physical measures including height and body mass index), sex (e.g., blood pressure related traits) and socioeconomic status (education). Our study represents the first comprehensive phenome-wide heritability analysis in the UK Biobank, and underscores the importance of considering population characteristics in interpreting heritability.

Ge, T., et al. Multidimensional heritability analysis of neuroanatomical shape. Nature Communications 7, 13291 (2016).Abstract

In the dawning era of large-scale biomedical data, multidimensional phenotype vectors will play an increasing role in examining the genetic underpinnings of brain features, behaviour and disease. For example, shape measurements derived from brain MRI scans are multidimensional geometric descriptions of brain structure and provide an alternate class of phenotypes that remains largely unexplored in genetic studies. Here we extend the concept of heritability to multidimensional traits, and present the first comprehensive analysis of the heritability of neuroanatomical shape measurements across an ensemble of brain structures based on genome-wide SNP and MRI data from 1,320 unrelated, young and healthy individuals. We replicate our findings in an extended twin sample from the Human Connectome Project (HCP). Our results demonstrate that neuroanatomical shape can be significantly heritable, above and beyond volume, and can serve as a complementary phenotype to study the genetic determinants and clinical relevance of brain structure.

Sabuncu, M.R., et al. Morphometricity as a measure of the neuroanatomical signature of a trait. Proceedings of the National Academy of Sciences 113, 39, E5749–E5756 (2016).Abstract

Complex physiological and behavioral traits, including neurological and psychiatric disorders, often associate with distributed anatomical variation. This paper introduces a global metric, called morphometricity, as a measure of the anatomical signature of different traits. Morphometricity is defined as the proportion of phenotypic variation that can be explained by macroscopic brain morphology. We estimate morphometricity via a linear mixed-effects model that uses an anatomical similarity matrix computed based on measurements derived from structural brain MRI scans. We examined over 3,800 unique MRI scans from nine large-scale studies to estimate the morphometricity of a range of phenotypes, including clinical diagnoses such as Alzheimers disease, and non-clinical traits such as measures of cognition. Our results demonstrate that morphometricity can provide novel insights about the neuroanatomical correlates of a diverse set of traits, revealing associations that might not be detectable through traditional statistical techniques. 

Krienen, F.M., Yeo, B.T.T., Ge, T., Buckner, R.L. & Sherwood, C.C. Transcriptional profiles of supragranular-enriched genes associate with corticocortical network architecture in the human brain. Proceedings of the National Academy of Sciences 113, 4, E469-E478 (2016).Abstract

The human brain is patterned with disproportionately large, distributed cerebral networks that connect multiple association zones in the frontal, temporal, and parietal lobes. The expansion of the cortical surface, along with the emergence of long-range connectivity networks, may be reflected in changes to the underlying molecular architecture. Using the Allen Institutes human brain transcriptional atlas, we demonstrate that genes particularly enriched in supragranular layers of the human cerebral cortex relative to mouse distinguish major cortical classes. The topography of transcriptional expression reflects large-scale brain network organization consistent with estimates from functional connectivity MRI and anatomical tracing in nonhuman primates. Microarray expression data for genes preferentially expressed in human upper layers (II/III), but enriched only in lower layers (V/VI) of mouse, were cross-correlated to identify molecular profiles across the cerebral cortex of postmortem human brains (n = 6). Unimodal sensory and motor zones have similar molecular profiles, despite being distributed across the cortical mantle. Sensory/motor profiles were anticorrelated with paralimbic and certain distributed association network profiles. Tests of alternative gene sets did not consistently distinguish sensory and motor regions from paralimbic and association regions: (i) genes enriched in supragranular layers in both humans and mice, (ii) genes cortically enriched in humans relative to nonhuman primates, (iii) genes related to connectivity in rodents, (iv) genes associated with human and mouse connectivity, and (v) 1,454 gene sets curated from known gene ontologies. Molecular innovations of upper cortical layers may be an important component in the evolution of long-range corticocortical projections. 

Ge, T., et al. Massively Expedited Genome-wide Heritability Analysis (MEGHA). Proceedings of the National Academy of Sciences 112, 8, 2479-2484 (2015).Abstract

The discovery and prioritization of heritable phenotypes is a computational challenge in a variety of settings, including neuroimaging genetics and analyses of the vast phenotypic repositories in electronic health record systems and population-based biobanks. Classical estimates of heritability require twin or pedigree data, which can be costly and difficult to acquire. Genome-wide complex trait analysis is an alternative tool to compute heritability estimates from unrelated individuals, using genome-wide data that are increasingly ubiquitous, but is computationally demanding and becomes difficult to apply in evaluating very large numbers of phenotypes. Here we present a fast and accurate statistical method for high-dimensional heritability analysis using genome-wide SNP data from unrelated individuals, termed massively expedited genome-wide heritability analysis (MEGHA) and accompanying nonparametric sampling techniques that enable flexible inferences for arbitrary statistics of interest. MEGHA produces estimates and significance measures of heritability with several orders of magnitude less computational time than existing methods, making heritability-based prioritization of millions of phenotypes based on data from unrelated individuals tractable for the first time to our knowledge. As a demonstration of application, we conducted heritability analyses on global and local morphometric measurements derived from brain structural MRI scans, using genome-wide SNP data from 1,320 unrelated young healthy adults of non-Hispanic European ancestry. We also computed surface maps of heritability for cortical thickness measures and empirically localized cortical regions where thickness measures were significantly heritable. Our analyses demonstrate the unique capability of MEGHA for large-scale heritability-based screening and high-dimensional heritability profile construction. 

Ge, T., et al. A kernel machine method for detecting effects of interaction between multidimensional variable sets: An imaging genetics application. NeuroImage 109, 505-514 (2015).Abstract

Measurements derived from neuroimaging data can serve as markers of disease and/or healthy development, are largely heritable, and have been increasingly utilized as (intermediate) phenotypes in genetic association studies. To date, imaging genetic studies have mostly focused on discovering isolated genetic effects, typically ignoring potential interactions with non-genetic variables such as disease risk factors, environmental exposures, and epigenetic markers. However, identifying significant interaction effects is critical for revealing the true relationship between genetic and phenotypic variables, and shedding light on disease mechanisms. In this paper, we present a general kernel machine based method for detecting effects of the interaction between multidimensional variable sets. This method can model the joint and epistatic effect of a collection of single nucleotide polymorphisms (SNPs), accommodate multiple factors that potentially moderate genetic influences, and test for nonlinear interactions between sets of variables in a flexible framework. As a demonstration of application, we applied the method to the data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) to detect the effects of the interactions between candidate Alzheimer's disease (AD) risk genes and a collection of cardiovascular disease (CVD) risk factors, on hippocampal volume measurements derived from structural brain magnetic resonance imaging (MRI) scans. Our method identified that two genes, CR1 and EPHA1, demonstrate significant interactions with CVD risk factors on hippocampal volume, suggesting that CR1 and EPHA1 may play a role in influencing AD-related neurodegeneration in the presence of CVD risks. 

Ge, T., Müller-Lenke, N., Bendfeldt, K., Nichols, T.E. & Johnson, T.D. Analysis of multiple sclerosis lesions via spatially varying coefficients. The Annals of Applied Statistics 8, 2, 1095-1118 (2014).Abstract

Magnetic resonance imaging (MRI) plays a vital role in the scientific investigation and clinical management of multiple sclerosis. Analyses of binary multiple sclerosis lesion maps are typically "mass univariate" and conducted with standard linear models that are ill suited to the binary nature of the data and ignore the spatial dependence between nearby voxels (volume elements). Smoothing the lesion maps does not entirely eliminate the non-Gaussian nature of the data and requires an arbitrary choice of the smoothing parameter. Here we present a Bayesian spatial model to accurately model binary lesion maps and to determine if there is spatial dependence between lesion location and subject specific covariates such as MS subtype, age, gender, disease duration and disease severity measures. We apply our model to binary lesion maps derived from T2-weighted MRI images from 250 multiple sclerosis patients classified into five clinical subtypes, and demonstrate unique modeling and predictive capabilities over existing methods. 

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Ge, T., Tian, X., Kurths, J., Feng, J. & Lin, W. Achieving modulated oscillations by feedback control. Physical Review E 90, 2, 022909 (2014).Abstract

In this paper, we develop an approach to achieve either frequency or amplitude modulation of an oscillator merely through feedback control. We present and implement a unified theory of our approach for any finite-dimensional continuous dynamical system that exhibits oscillatory behavior. The approach is illustrated not only for the normal forms of dynamical systems but also for representative biological models, such as the isolated and coupled FitzHugh-Nagumo model. We demonstrate the potential usefulness of our approach to uncover the mechanisms of frequency and amplitude modulations experimentally observed in a wide range of real systems. 

Thompson, P.M., Ge, T., Glahn, D.C., Jahanshad, N. & Nichols, T.E. Genetics of the connectome. NeuroImage 80, 475-488 (2013).Abstract

Connectome genetics attempts to discover how genetic factors affect brain connectivity. Here we review a variety of genetic analysis methods — such as genome-wide association studies (GWAS), linkage and candidate gene studies — that have been fruitfully adapted to imaging data to implicate specific variants in the genome for brain-related traits. Studies that emphasized the genetic influences on brain connectivity. Some of these analyses of brain integrity and connectivity using diffusion MRI, and others have mapped genetic effects on functional networks using resting state functional MRI. Connectome-wide genome-wide scans have also been conducted, and we review the multivariate methods required to handle the extremely high dimension of the genomic and network data. We also review some consortium efforts, such as ENIGMA, that offer the power to detect robust common genetic associations using phenotypic harmonization procedures and meta-analysis. Current work on connectome genetics is advancing on many fronts and promises to shed light on how disease risk genes affect the brain. It is already discovering new genetic loci and even entire genetic networks that affect brain organization and connectivity. 

Luo, Q., Ge, T., Grabenhorst, F., Feng, J. & Rolls, E.T. Attention-dependent modulation of cortical taste circuits revealed by Granger causality with signal-dependent noise. PLoS Computational Biology 9, 10, e1003265 (2013).Abstract

We show, for the first time, that in cortical areas, for example the insular, orbitofrontal, and lateral prefrontal cortex, there is signal-dependent noise in the fMRI blood-oxygen level dependent (BOLD) time series, with the variance of the noise increasing approximately linearly with the square of the signal. Classical Granger causal models are based on autoregressive models with time invariant covariance structure, and thus do not take this signal-dependent noise into account. To address this limitation, here we describe a Granger causal model with signal-dependent noise, and a novel, likelihood ratio test for causal inferences. We apply this approach to the data from an fMRI study to investigate the source of the top-down attentional control of taste intensity and taste pleasantness processing. The Granger causality with signal-dependent noise analysis reveals effects not identified by classical Granger causal analysis. In particular, there is a top-down effect from the posterior lateral prefrontal cortex to the insular taste cortex during attention to intensity but not to pleasantness, and there is a top-down effect from the anterior and posterior lateral prefrontal cortex to the orbitofrontal cortex during attention to pleasantness but not to intensity. In addition, there is stronger forward effective connectivity from the insular taste cortex to the orbitofrontal cortex during attention to pleasantness than during attention to intensity. These findings indicate the importance of explicitly modeling signal-dependent noise in functional neuroimaging, and reveal some of the processes involved in a biased activation theory of selective attention.

Ge, T., Feng, J., Hibar, D.P., Thompson, P.M. & Nichols, T.E. Increasing power for voxel-wise genome-wide association studies: The random field theory, least square kernel machines and fast permutation procedures. NeuroImage 63, 858-873 (2012).Abstract

Imaging traits are thought to have more direct links to genetic variation than diagnostic measures based on cognitive or clinical assessments and provide a powerful substrate to examine the influence of genetics on human brains. Although imaging genetics has attracted growing attention and interest, most brain-wide genome-wide association studies focus on voxel-wise single-locus approaches, without taking advantage of the spatial information in images or combining the effect of multiple genetic variants. In this paper we present a fast implementation of voxel- and cluster-wise inferences based on the random field theory to fully use the spatial information in images. The approach is combined with a multi-locus model based on least square kernel machines to associate the joint effect of several single nucleotide polymorphisms (SNP) with imaging traits. A fast permutation procedure is also proposed which significantly reduces the number of permutations needed relative to the standard empirical method and provides accurate small p-value estimates based on parametric tail approximation. We explored the relation between 448,294 single nucleotide polymorphisms and 18,043 genes in 31,662 voxels of the entire brain across 740 elderly subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Structural MRI scans were analyzed using tensor-based morphometry (TBM) to compute 3D maps of regional brain volume differences compared to an average template image based on healthy elderly subjects. We find method to be more sensitive compared with voxel-wise single-locus approaches. A number of genes were identified as having significant associations with volumetric changes. The most associated gene was GRIN2B, which encodes the N-methyl-D-aspartate (NMDA) glutamate receptor NR2B subunit and affects both the parietal and temporal lobes in human brains. Its role in Alzheimer's disease has been widely acknowledged and studied, suggesting the validity of the approach. The various advantages over existing approaches indicate a great potential offered by this novel framework to detect genetic influences on human brains. 

Ge, T., Lin, W. & Feng, J. Invariance principles allowing of non-Lyapunov functions for estimating attractor boundaries of discrete dynamical systems. IEEE Transactions on Automatic Control 57, 2, 500-505 (2012).Abstract

This technical note establishes several versions of invariance principles for describing the eventual dynamical behaviors of discrete dynamical systems. Instead of the requirement of the so-called Lyapunov functions in the classical LaSalle invariance principle, some more relaxed conditions are imported. The established invariance principles thus can be applied to a more general class of discrete dynamical systems for classifying their orbits into two categories based on the eventual dynamical behaviors, and the proposed classification scheme is suitable for theoretically and numerically estimating the local or global attractors produced by the discrete dynamical systems. The practical usefulness of the analytical results is verified by systematically investigating several representative discrete systems.