Heritability of "small-world" networks in the brain: a graph theoretical analysis of resting-state EEG functional connectivity

Linking genes, circuits, and behavior: network connectivity as a novel endophenotype of externalizing

BACKGROUND

Externalizing disorders are known to be partly heritable, but the biological pathways linking genetic risk to the manifestation of these costly behaviors remain under investigation. This study sought to identify neural phenotypes associated with genomic vulnerability for externalizing disorders.

METHODS

One-hundred fifty-five White, non-Hispanic veterans were genotyped using a genome-wide array and underwent resting-state functional magnetic resonance imaging. Genetic susceptibility was assessed using an independently developed polygenic score (PS) for externalizing, and functional neural networks were identified using graph theory based network analysis. Tasks of inhibitory control and psychiatric diagnosis (alcohol/substance use disorders) were used to measure externalizing phenotypes.

RESULTS

A polygenic externalizing disorder score (PS) predicted connectivity in a brain circuit (10 nodes, nine links) centered on left amygdala that included several cortical [bilateral inferior frontal gyrus (IFG) pars triangularis, left rostral anterior cingulate cortex (rACC)] and subcortical (bilateral amygdala, hippocampus, and striatum) regions. Directional analyses revealed that bilateral amygdala influenced left prefrontal cortex (IFG) in participants scoring higher on the externalizing PS, whereas the opposite direction of influence was observed for those scoring lower on the PS. Polygenic variation was also associated with higher Participation Coefficient for bilateral amygdala and left rACC, suggesting that genes related to externalizing modulated the extent to which these nodes functioned as communication hubs.

CONCLUSIONS

Findings suggest that externalizing polygenic risk is associated with disrupted connectivity in a neural network implicated in emotion regulation, impulse control, and reinforcement learning. Results provide evidence that this network represents a genetically associated neurobiological vulnerability for externalizing disorders.

Uncovering the Transcriptional Correlates of Hub Connectivity in Neural Networks

Connections in nervous systems are disproportionately concentrated on a small subset of neural elements that act as network hubs. Hubs have been found across different species and scales ranging from C. elegans to mouse, rat, cat, macaque, and human, suggesting a role for genetic influences. The recent availability of brain-wide gene expression atlases provides new opportunities for mapping the transcriptional correlates of large-scale network-level phenotypes. Here we review studies that use these atlases to investigate gene expression patterns associated with hub connectivity in neural networks and present evidence that some of these patterns are conserved across species and scales.

Exact topological inference of the resting-state brain networks in twins

A cycle in a brain network is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. Whereas the number of connected components represents the integration of the brain network, the number of cycles represents how strong the integration is. However, it is unclear how to perform statistical inference on the number of cycles in the brain network. In this study, we present a new statistical inference framework for determining the significance of the number of cycles through the Kolmogorov-Smirnov (KS) distance, which was recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using the random network simulations with ground truths. By using a twin imaging study, which provides biological ground truth, the methods are applied in determining if the number of cycles is a statistically significant heritable network feature in the resting-state functional connectivity in 217 twins obtained from the Human Connectome Project. The MATLAB codes as well as the connectivity matrices used in generating results are provided at http://www.stat.wisc.edu/∼mchung/TDA.

In this paper, we propose a new topological distance based on the Kolmogorov-Smirnov (KS) distance that is adapted for brain networks, and compare them against other topological network distances including the Gromov-Hausdorff (GH) distances. KS-distance is recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number, which measures the number of connected components. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using random network simulations with ground truths. Using a twin imaging study, which provides biological ground truth (of network differences), we demonstrate that the KS distances on the zeroth and first Betti numbers have the ability to determine heritability.

Altered structural and functional brain network overall organization predict human intertemporal decision-making

Intertemporal decision-making is naturally ubiquitous to us: individuals always make a decision with different consequences occurring at different moments. These choices are invariably involved in life-changing outcomes regarding marriage, education, fertility, long-term well-being, and even public policy. Previous studies have clearly uncovered the neurobiological mechanism of the intertemporal decision in the schemes of regional location or sub-network. However, it still remains unclear how to characterize intertemporal behavior with multimodal whole-brain network metrics to date. Here, we combined diffusion tensor image and resting-state functional connectivity MRI technology, in conjunction with graph-theoretical analysis, to explore the link between topological properties of integrated structural and functional whole-brain networks and intertemporal decision-making. Graph-theoretical analysis illustrated that the participants with steep discounting rates exhibited the decreased global topological organizations including small-world and rich-club regimes in both functional and structural connectivity networks, and reflected the dreadful local topological dynamics in the modularity of functional connectome. Furthermore, in the cross-modalities configuration, the same relationship was predominantly observed for the coupling of structural-functional connectivity as well. Above topological metrics are commonly indicative of the communication pattern of simultaneous global and local parallel information processing, and it thus reshapes our accounts on intertemporal decision-making from functional regional/sub-network scheme to multimodal brain overall organization.

Altered Forebrain Functional Connectivity and Neurotransmission in a Kinase-Inactive Met Mouse Model of Autism

MET, the gene encoding the tyrosine kinase receptor for hepatocyte growth factor, is a susceptibility gene for autism spectrum disorder (ASD). Genetically altered mice with a kinase-inactive Met offer a potential model for understanding neural circuit organization changes in autism. Here, we focus on the somatosensory thalamocortical circuitry because distinct somatosensory sensitivity phenotypes accompany ASD, and this system plays a major role in sensorimotor and social behaviors in mice. We employed resting-state functional magnetic resonance imaging and in vivo high-resolution proton MR spectroscopy to examine neuronal connectivity and neurotransmission of wild-type, heterozygous Met-Emx1, and fully inactive homozygous Met-Emx1 mice. Met-Emx1 brains showed impaired maturation of large-scale somatosensory network connectivity when compared with wild-type controls. Significant sex × genotype interaction in both network features and glutamate/gamma-aminobutyric acid (GABA) balance was observed. Female Met-Emx1 brains showed significant connectivity and glutamate/GABA balance changes in the somatosensory thalamocortical system when compared with wild-type brains. The glutamate/GABA ratio in the thalamus was correlated with the connectivity between the somatosensory cortex and the thalamus in heterozygous Met-Emx1 female brains. The findings support the hypothesis that aberrant functioning of the somatosensory thalamocortical system is at the core of the conspicuous somatosensory behavioral phenotypes observed in Met-Emx1 mice.

Increased Functional Brain Network Efficiency During Audiovisual Temporal Asynchrony Integration Task in Aging

Audiovisual integration significantly changes over the lifespan, but age-related functional connectivity in audiovisual temporal asynchrony integration tasks remains underexplored. In the present study, electroencephalograms (EEGs) of 27 young adults (22–25 years) and 25 old adults (61–76 years) were recorded during an audiovisual temporal asynchrony integration task with seven conditions [auditory (A), visual (V), AV, A50V, A100V, V50A and V100A]. We calculated the phase lag index (PLI)-weighted connectivity networks modulated by the audiovisual tasks and found that the PLI connections showed obvious dynamic changes after stimulus onset. In the theta (4–7 Hz) and alpha (8–13 Hz) bands, the AV and V50A conditions induced stronger functional connections and higher global and local efficiencies, reflecting a stronger audiovisual integration effect, which was attributed to the auditory information arriving at the primary auditory cortex earlier than the visual information reaching the primary visual cortex. Importantly, the functional connectivity and network efficiencies of old adults revealed higher global and local efficiencies and higher degree in both the theta and alpha bands. These larger network efficiencies indicated that old adults might experience more difficulties in attention and cognitive control during the audiovisual integration task with temporal asynchrony than young adults. There were significant associations between network efficiencies and peak time of integration only in young adults. We propose that an audiovisual task with multiple conditions might arouse the appropriate attention in young adults but would lead to a ceiling effect in old adults. Our findings provide new insights into the network topography of old adults during audiovisual integration and highlight higher functional connectivity and network efficiencies due to greater cognitive demand.

Variability and stability of large-scale cortical oscillation patterns

Individual differences in brain organization exist at many spatiotemporal scales and underlie the diversity of human thought and behavior. Oscillatory neural activity is crucial for these processes, but how such rhythms are expressed across the cortex within and across individuals is poorly understood. We conducted a systematic characterization of brain-wide activity across frequency bands and oscillatory features during rest and task execution. We found that oscillatory profiles exhibit sizable group-level similarities, indicating the presence of common templates of oscillatory organization. Nonetheless, well-defined subject-specific network profiles were discernible beyond the structure shared across individuals. These individualized patterns were sufficiently stable to recognize individuals several months later. Moreover, network structure of rhythmic activity varied considerably across distinct oscillatory frequencies and features, indicating the existence of several parallel information processing streams embedded in distributed electrophysiological activity. These findings suggest that network similarity analyses may be useful for understanding the role of large-scale brain oscillations in physiology and behavior.

Neural oscillations are critical for the human brain’s ability to optimally respond to complex environmental input. However, relatively little is known about the network properties of these oscillatory rhythms. We used electroencephalography (EEG) to analyze large-scale brain wave patterns, focusing on multiple frequency bands and several key features of oscillatory communication. We show that networks defined in this manner are, in fact, distinct, suggesting that EEG activity encompasses multiple, parallel information processing streams. Remarkably, the same networks can be used to uniquely identify individuals over a period of approximately half a year, thus serving as neural fingerprints. These findings indicate that investigating oscillatory dynamics from a network perspective holds considerable promise as a tool to understand human cognition and behavior.

Dynamic Information Flow Based on EEG and Diffusion MRI in Stroke: A Proof-of-Principle Study

In hemiparetic stroke, functional recovery of paretic limb may occur with the reorganization of neural networks in the brain. Neuroimaging techniques, such as magnetic resonance imaging (MRI), have a high spatial resolution which can be used to reveal anatomical changes in the brain following a stroke. However, low temporal resolution of MRI provides less insight of dynamic changes of brain activity. In contrast, electro-neurophysiological techniques, such as electroencephalography (EEG), have an excellent temporal resolution to measure such transient events, however are hindered by its low spatial resolution. This proof-of-principle study assessed a novel multimodal brain imaging technique namely Variational Bayesian Multimodal Encephalography (VBMEG), which aims to improve the spatial resolution of EEG for tracking the information flow inside the brain and its changes following a stroke. The limitations of EEG are complemented by constraints derived from anatomical MRI and diffusion weighted imaging (DWI). EEG data were acquired from individuals suffering from a stroke as well as able-bodied participants while electrical stimuli were delivered sequentially at their index finger in the left and right hand, respectively. The locations of active sources related to this stimulus were precisely identified, resulting in high Variance Accounted For (VAF above 80%). An accurate estimation of dynamic information flow between sources was achieved in this study, showing a high VAF (above 90%) in the cross-validation test. The estimated dynamic information flow was compared between chronic hemiparetic stroke and able-bodied individuals. The results demonstrate the feasibility of VBMEG method in revealing the changes of information flow in the brain after stroke. This study verified the VBMEG method as an advanced computational approach to track the dynamic information flow in the brain following a stroke. This may lead to the development of a quantitative tool for monitoring functional changes of the cortical neural networks after a unilateral brain injury and therefore facilitate the research into, and the practice of stroke rehabilitation.

Investigation Of Interaction Between Physiological Signals And fMRI Dynamic Functional Connectivity Using Independent Component Analysis

The blood oxygen level dependent (BOLD) fMRI signal is influenced not only by neuronal activity but also by fluctuations in physiological signals, including respiration, arterial CO2 and heart rate/ heart rate variability (HR/HRV). Even spontaneous physiological signal fluctuations have been shown to influence the BOLD fMRI signal in a regionally specific manner. Consequently, estimates of functional connectivity between different brain regions, performed when the subject is at rest, may be confounded by the effects of physiological signal fluctuations. In addition, resting functional connectivity has been shown to vary with respect to time (dynamic functional connectivity - DFC), with the sources of this variation not fully elucidated. The effect of physiological factors on dynamic (time-varying) resting-state functional connectivity has not been studied extensively, to our knowledge. In our previous study, we investigated the effect of heart rate (HR) and end-tidal CO2 (PETCO2) on the time-varying network degree of three well-described RSNs (DMN, SMN and Visual Network) using mask-based and seed-based analysis, and we identified brain-heart interactions which were more pronounced in specific frequency bands. Here, we extend this work, by estimating DFC and its corresponding network degree for the RSNs, employing a data-driven approach to extract the RSNs (low-and high-dimensional Independent Component Analysis (ICA)), which we subsequently correlate with the characteristics of simultaneously collected physiological signals. The results confirm that physiological signals have a modulatory effect on resting-state, fMRI-based DFC.

Feature Extraction with GMDH-Type Neural Networks for EEG-Based Person Identification

The brain activity observed on EEG electrodes is influenced by volume conduction and functional connectivity of a person performing a task. When the task is a biometric test the EEG signals represent the unique “brain print”, which is defined by the functional connectivity that is represented by the interactions between electrodes, whilst the conduction components cause trivial correlations. Orthogonalization using autoregressive modeling minimizes the conduction components, and then the residuals are related to features correlated with the functional connectivity. However, the orthogonalization can be unreliable for high-dimensional EEG data. We have found that the dimensionality can be significantly reduced if the baselines required for estimating the residuals can be modeled by using relevant electrodes. In our approach, the required models are learnt by a Group Method of Data Handling (GMDH) algorithm which we have made capable of discovering reliable models from multidimensional EEG data. In our experiments on the EEG-MMI benchmark data which include 109 participants, the proposed method has correctly identified all the subjects and provided a statistically significant ([Formula: see text]) improvement of the identification accuracy. The experiments have shown that the proposed GMDH method can learn new features from multi-electrode EEG data, which are capable to improve the accuracy of biometric identification.

Applications of community detection techniques to brain graphs: Algorithmic considerations and implications for neural function

The human brain can be represented as a graph in which neural units such as cells or small volumes of tissue are heterogeneously connected to one another through structural or functional links. Brain graphs are parsimonious representations of neural systems that have begun to offer fundamental insights into healthy human cognition, as well as its alteration in disease. A critical open question in network neuroscience lies in how neural units cluster into densely interconnected groups that can provide the coordinated activity that is characteristic of perception, action, and adaptive behaviors. Tools that have proven particularly useful for addressing this question are community detection approaches, which can identify communities or modules: groups of neural units that are densely interconnected with other units in their own group but sparsely interconnected with units in other groups. In this paper, we describe a common community detection algorithm known as modularity maximization, and we detail its applications to brain graphs constructed from neuroimaging data. We pay particular attention to important algorithmic considerations, especially in recent extensions of these techniques to graphs that evolve in time. After recounting a few fundamental insights that these techniques have provided into brain function, we highlight potential avenues of methodological advancements for future studies seeking to better characterize the patterns of coordinated activity in the brain that accompany human behavior. This tutorial provides a naive reader with an introduction to theoretical considerations pertinent to the generation of brain graphs, an understanding of modularity maximization for community detection, a resource of statistical measures that can be used to characterize community structure, and an appreciation of the usefulness of these approaches in uncovering behaviorally-relevant network dynamics in neuroimaging data.

Transcranial direct current stimulation generates a transient increase of small-world in brain connectivity: an EEG graph theoretical analysis

Transcranial direct current stimulation (tDCS) is a non-invasive technique able to modulate cortical excitability in a polarity-dependent way. At present, only few studies investigated the effects of tDCS on the modulation of functional connectivity between remote cortical areas. The aim of this study was to investigate-through graph theory analysis-how bipolar tDCS modulate cortical networks high-density EEG recordings were acquired before and after bipolar cathodal, anodal and sham tDCS involving the primary motor and pre-motor cortices of the dominant hemispherein 14 healthy subjects. Results showed that, after bipolar anodal tDCS stimulation, brain networks presented a less evident "small world" organization with a global tendency to be more random in its functional connections with respect to prestimulus condition in both hemispheres. Results suggest that tDCS globally modulates the cortical connectivity of the brain, modifying the underlying functional organization of the stimulated networks, which might be related to changes in synaptic efficiency of the motor network and related brain areas. This study demonstrated that graph analysis approach to EEG recordings is able to intercept changes in cortical functions mediated by bipolar anodal tDCS mainly involving the dominant M1 and related motor areas. Concluding, tDCS could be an useful technique to help understanding brain rhythms and their topographic functional organization and specificity.

Network Properties in Transitions of Consciousness during Propofol-induced Sedation

Reliable electroencephalography (EEG) signatures of transitions between consciousness and unconsciousness under anaesthesia have not yet been identified. Herein we examined network changes using graph theoretical analysis of high-density EEG during patient-titrated propofol-induced sedation. Responsiveness was used as a surrogate for consciousness. We divided the data into five states: baseline, transition into unresponsiveness, unresponsiveness, transition into responsiveness, and recovery. Power spectral analysis showed that delta power increased from responsiveness to unresponsiveness. In unresponsiveness, delta waves propagated from frontal to parietal regions as a traveling wave. Local increases in delta connectivity were evident in parietal but not frontal regions. Graph theory analysis showed that increased local efficiency could differentiate the levels of responsiveness. Interestingly, during transitions of responsive states, increased beta connectivity was noted relative to consciousness and unconsciousness, again with increased local efficiency. Abrupt network changes are evident in the transitions in responsiveness, with increased beta band power/connectivity marking transitions between responsive states, while the delta power/connectivity changes were consistent with the fading of consciousness using its surrogate responsiveness. These results provide novel insights into the neural correlates of these behavioural transitions and EEG signatures for monitoring the levels of consciousness under sedation.

Effect of Field Spread on Resting-State Magneto Encephalography Functional Network Analysis: A Computational Modeling Study

A popular way to analyze resting-state electroencephalography (EEG) and magneto encephalography (MEG) data is to treat them as a functional network in which sensors are identified with nodes and the interaction between channel time series and the network connections. Although conceptually appealing, the network-theoretical approach to sensor-level EEG and MEG data is challenged by the fact that EEG and MEG time series are mixtures of source activity. It is, therefore, of interest to assess the relationship between functional networks of source activity and the ensuing sensor-level networks. Since these topological features are of high interest in experimental studies, we address the question of to what extent the network topology can be reconstructed from sensor-level functional connectivity (FC) measures in case of MEG data. Simple simulations that consider only a small number of regions do not allow to assess network properties; therefore, we use a diffusion magnetic resonance imaging-constrained whole-brain computational model of resting-state activity. Our motivation lies behind the fact that still many contributions found in the literature perform network analysis at sensor level, and we aim at showing the discrepancies between source- and sensor-level network topologies by using realistic simulations of resting-state cortical activity. Our main findings are that the effect of field spread on network topology depends on the type of interaction (instantaneous or lagged) and leads to an underestimation of lagged FC at sensor level due to instantaneous mixing of cortical signals, instantaneous interaction is more sensitive to field spread than lagged interaction, and discrepancies are reduced when using planar gradiometers rather than axial gradiometers. We, therefore, recommend using lagged interaction measures on planar gradiometer data when investigating network properties of resting-state sensor-level MEG data.

Connectivity maps based analysis of EEG for the advanced diagnosis of schizophrenia attributes

This article presents a novel connectivity analysis method that is suitable for multi-node networks such as EEG, MEG or EcOG electrode recordings. Its diagnostic power and ability to interpret brain states in schizophrenia is demonstrated on a set of 50 subjects that constituted of 25 healthy and 25 diagnosed with schizophrenia and treated with medication. The method can also be used for the automatic detection of schizophrenia; it exhibits higher sensitivity than state-of-the-art methods with no false positives. The detection is based on an analysis from a minute long pattern-recognition computer task. Moreover, this connectivity analysis leads naturally to an optimal choice of electrodes and hence to highly statistically significant results that are based on data from only 3-5 electrodes. The method is general and can be used for the diagnosis of other psychiatric conditions, provided an appropriate computer task is devised.

Functional and effective whole brain connectivity using magnetoencephalography to identify monozygotic twin pairs

Resting-state functional connectivity patterns are highly stable over time within subjects. This suggests that such 'functional fingerprints' may have strong genetic component. We investigated whether the functional (FC) or effective (EC) connectivity patterns of one monozygotic twin could be used to identify the co-twin among a larger sample and determined the overlap in functional fingerprints within monozygotic (MZ) twin pairs using resting state magnetoencephalography (MEG). We included 32 cognitively normal MZ twin pairs from the Netherlands Twin Register who participate in the EMIF-AD preclinAD study (average age 68 years). Combining EC information across multiple frequency bands we obtained an identification rate over 75%. Since MZ twin pairs are genetically identical these results suggest a high genetic contribution to MEG-based EC patterns, leading to large similarities in brain connectivity patterns between two individuals even after 60 years of life or more.

Disrupted resting brain graph measures in individuals at high risk for alcoholism

Familial susceptibility to alcoholism is likely to be linked to the externalizing diathesis seen in high-risk offspring from high-density alcohol use disorder (AUD) families. The present study aimed at comparing resting brain functional connectivity and their association with externalizing symptoms and alcoholism familial density in 40 substance-naive high-risk (HR) male offspring from high-density AUD families and 30 matched healthy low-risk (LR) males without a family history of substance dependence using graph theory-based network analysis. The HR subjects from high-density AUD families compared with LR, showed significantly reduced clustering, small-worldness, and local network efficiency. The frontoparietal, cingulo-opercular, sensorimotor and cerebellar networks exhibited significantly reduced functional segregation. These disruptions exhibited independent incremental value in predicting the externalizing symptoms over and above the demographic variables. The reduction of functional segregation in HR subjects was significant across both the younger and older age groups and was proportional to the family loading of AUDs. Detection and estimation of these developmentally relevant disruptions in small-world architecture at critical brain regions sub-serving cognitive, affective, and sensorimotor processes are vital for understanding the familial risk for early onset alcoholism as well as for understanding the pathophysiological mechanism of externalizing behaviors.

The heritability of multi-modal connectivity in human brain activity

Patterns of intrinsic human brain activity exhibit a profile of functional connectivity that is associated with behaviour and cognitive performance, and deteriorates with disease. This paper investigates the relative importance of genetic factors and the common environment between twins in determining this functional connectivity profile. Using functional magnetic resonance imaging (fMRI) on 820 subjects from the Human Connectome Project, and magnetoencephalographic (MEG) recordings from a subset, the heritability of connectivity among 39 cortical regions was estimated. On average over all connections, genes account for about 15% of the observed variance in fMRI connectivity (and about 10% in alpha-band and 20% in beta-band oscillatory power synchronisation), which substantially exceeds the contribution from the environment shared between twins. Therefore, insofar as twins share a common upbringing, it appears that genes, rather than the developmental environment, have the dominant role in determining the coupling of neuronal activity.

EEG analysis of seizure patterns using visibility graphs for detection of generalized seizures

BACKGROUND

The traditional EEG features in the time and frequency domain show limited seizure detection performance in the epileptic population with intellectual disability (ID). In addition, the influence of EEG seizure patterns on detection performance was less studied.

NEW METHOD

A single-channel EEG signal can be mapped into visibility graphs (VGS), including basic visibility graph (VG), horizontal VG (HVG), and difference VG (DVG). These graphs were used to characterize different EEG seizure patterns. To demonstrate its effectiveness in identifying EEG seizure patterns and detecting generalized seizures, EEG recordings of 615h on one EEG channel from 29 epileptic patients with ID were analyzed.

RESULTS

A novel feature set with discriminative power for seizure detection was obtained by using the VGS method. The degree distributions (DDs) of DVG can clearly distinguish EEG of each seizure pattern. The degree entropy and power-law degree power in DVG were proposed here for the first time, and they show significant difference between seizure and non-seizure EEG. The connecting structure measured by HVG can better distinguish seizure EEG from background than those by VG and DVG.

COMPARISON WITH EXISTING METHOD

A traditional EEG feature set based on frequency analysis was used here as a benchmark feature set. With a support vector machine (SVM) classifier, the seizure detection performance of the benchmark feature set (sensitivity of 24%, FD_t/h of 1.8s) can be improved by combining our proposed VGS features extracted from one EEG channel (sensitivity of 38%, FD_t/h of 1.4s).

CONCLUSIONS

The proposed VGS-based features can help improve seizure detection for ID patients.

Detecting large-scale networks in the human brain using high-density electroencephalography

High-density electroencephalography (hdEEG) is an emerging brain imaging technique that can be used to investigate fast dynamics of electrical activity in the healthy and the diseased human brain. Its applications are however currently limited by a number of methodological issues, among which the difficulty in obtaining accurate source localizations. In particular, these issues have so far prevented EEG studies from reporting brain networks similar to those previously detected by functional magnetic resonance imaging (fMRI). Here, we report for the first time a robust detection of brain networks from resting state (256-channel) hdEEG recordings. Specifically, we obtained 14 networks previously described in fMRI studies by means of realistic 12-layer head models and exact low-resolution brain electromagnetic tomography (eLORETA) source localization, together with independent component analysis (ICA) for functional connectivity analysis. Our analyses revealed three important methodological aspects. First, brain network reconstruction can be improved by performing source localization using the gray matter as source space, instead of the whole brain. Second, conducting EEG connectivity analyses in individual space rather than on concatenated datasets may be preferable, as it permits to incorporate realistic information on head modeling and electrode positioning. Third, the use of a wide frequency band leads to an unbiased and generally accurate reconstruction of several network maps, whereas filtering data in a narrow frequency band may enhance the detection of specific networks and penalize that of others. We hope that our methodological work will contribute to rise of hdEEG as a powerful tool for brain research. Hum Brain Mapp 38:4631-4643, 2017. © 2017 Wiley Periodicals, Inc.

Functional Dysconnection of the Inferior Frontal Gyrus in Young People With Bipolar Disorder or at Genetic High Risk

BACKGROUND

Bipolar disorder (BD) is characterized by a dysregulation of affect and impaired integration of emotion with cognition. These traits are also expressed in probands at high genetic risk of BD. The inferior frontal gyrus (IFG) is a key cortical hub in the circuits of emotion and cognitive control, and it has been frequently associated with BD. Here, we studied resting-state functional connectivity of the left IFG in participants with BD and in those at increased genetic risk.

METHODS

Using resting-state functional magnetic resonance imaging we compared 49 young BD participants, 71 individuals with at least one first-degree relative with BD (at-risk), and 80 control subjects. We performed between-group analyses of the functional connectivity of the left IFG and used graph theory to study its local functional network topology. We also used machine learning to study classification based solely on the functional connectivity of the IFG.

RESULTS

In BD, the left IFG was functionally dysconnected from a network of regions, including bilateral insulae, ventrolateral prefrontal gyri, superior temporal gyri, and the putamen (p < .001). A small network incorporating neighboring insular regions and the anterior cingulate cortex showed weaker functional connectivity in at-risk than control participants (p < .006). These constellations of regions overlapped with frontolimbic regions that a machine learning classifier selected as predicting group membership with an accuracy significantly greater than chance.

CONCLUSIONS

Functional dysconnectivity of the IFG from regions involved in emotional regulation may represent a trait abnormality for BD and could potentially aid clinical diagnosis.

Small-world human brain networks: Perspectives and challenges

Modelling the human brain as a complex network has provided a powerful mathematical framework to characterize the structural and functional architectures of the brain. In the past decade, the combination of non-invasive neuroimaging techniques and graph theoretical approaches enable us to map human structural and functional connectivity patterns (i.e., connectome) at the macroscopic level. One of the most influential findings is that human brain networks exhibit prominent small-world organization. Such a network architecture in the human brain facilitates efficient information segregation and integration at low wiring and energy costs, which presumably results from natural selection under the pressure of a cost-efficiency balance. Moreover, the small-world organization undergoes continuous changes during normal development and ageing and exhibits dramatic alterations in neurological and psychiatric disorders. In this review, we survey recent advances regarding the small-world architecture in human brain networks and highlight the potential implications and applications in multidisciplinary fields, including cognitive neuroscience, medicine and engineering. Finally, we highlight several challenging issues and areas for future research in this rapidly growing field.

Can neurophysiologic measures serve as biomarkers for the efficacy of repetitive transcranial magnetic stimulation treatment of major depressive disorder?

Repetitive transcranial magnetic stimulation (rTMS) is an effective treatment for Major Depressive Disorder (MDD). There are clinical data that support the efficacy of many different approaches to rTMS treatment, and it remains unclear what combination of stimulation parameters is optimal to relieve depressive symptoms. Because of the costs and complexity of studies that would be necessary to explore and compare the large number of combinations of rTMS treatment parameters, it would be useful to establish reliable surrogate biomarkers of treatment efficacy that could be used to compare different approaches to treatment. This study reviews the evidence that neurophysiologic measures of cortical excitability could be used as biomarkers for screening different rTMS treatment paradigms. It examines evidence that: (1) changes in excitability are related to the mechanism of action of rTMS; (2) rTMS has consistent effects on measures of excitability that could constitute reliable biomarkers; and (3) changes in excitability are related to the outcomes of rTMS treatment of MDD. An increasing body of evidence indicates that these neurophysiologic measures have the potential to serve as reliable biomarkers for screening different approaches to rTMS treatment of MDD.

Altered Topological Properties of Brain Networks in Social Anxiety Disorder: A Resting-state Functional MRI Study

Recent studies involving connectome analysis including graph theory have yielded potential biomarkers for mental disorders. In this study, we aimed to investigate the differences of resting-state network between patients with social anxiety disorder (SAD) and healthy controls (HCs), as well as to distinguish between individual subjects using topological properties. In total, 42 SAD patients and the same number of HCs underwent resting functional MRI, and the topological organization of the whole-brain functional network was calculated using graph theory. Compared with the controls, the patients showed a decrease in 49 positive connections. In the topological analysis, the patients showed an increase in the area under the curve (AUC) of the global shortest path length of the network (L_p) and a decrease in the AUC of the global clustering coefficient of the network (C_p). Furthermore, the AUCs of L_p and C_p were used to effectively discriminate the individual SAD patients from the HCs with high accuracy. This study revealed that the neural networks of the SAD patients showed changes in topological characteristics, and these changes were prominent not only in both groups but also at the individual level. This study provides a new perspective for the identification of patients with SAD.

Variation at NRG1 genotype related to modulation of small-world properties of the functional cortical network

Functional brain networks possess significant small-world (SW) properties. Genetic variation relevant to both inhibitory and excitatory transmission may contribute to modulate these properties. In healthy controls, genotypic variation in Neuregulin 1 (NRG1) related to the risk of psychosis (risk alleles) would contribute to functional SW modulation of the cortical network. Electroencephalographic activity during an odd-ball task was recorded in 144 healthy controls. Then, small-worldness (SWn) was calculated in five frequency bands (i.e., theta, alpha, beta1, beta2 and gamma) for baseline (from -300 to the stimulus onset) and response (150-450 ms post-target stimulus) windows. The SWn modulation was defined as the difference in SWn between both windows. Association between SWn modulation and carrying the risk allele for three single nucleotide polymorphisms (SNP) of NRG1 (i.e., rs6468119, rs6994992 and rs7005606) was assessed. A significant association between three SNPs of NRG1 and the SWn modulation was found, specifically: NRG1 rs6468119 in alpha and beta1 bands; NRG1 rs6994992 in theta band; and NRG1 rs7005606 in theta and beta1 bands. Genetic variation at NRG1 may influence functional brain connectivity through the modulation of SWn properties of the cortical network.

Correspondence between evoked and intrinsic functional brain network configurations

Much of the literature exploring differences between intrinsic and task-evoked brain architectures has examined changes in functional connectivity patterns between specific brain regions. While informative, this approach overlooks important overall functional changes in hub organization and network topology that may provide insights about differences in integration between intrinsic and task-evoked states. Examination of changes in overall network organization, such as a change in the concentration of hub nodes or a quantitative change in network organization, is important for understanding the underlying processes that differ between intrinsic and task-evoked brain architectures. The present study used graph-theoretical techniques applied to publicly available neuroimaging data collected from a large sample of individuals (N = 202), and a within-subject design where resting-state and several task scans were collected from each participant as part of the Human Connectome Project. We demonstrate that differences between intrinsic and task-evoked brain networks are characterized by a task-general shift in high-connectivity hubs from primarily sensorimotor/auditory processing areas during the intrinsic state to executive control/salience network areas during task performance. In addition, we demonstrate that differences between intrinsic and task-evoked architectures are associated with changes in overall network organization, such as increases in network clustering, global efficiency and integration between modules. These findings offer a new perspective on the principles guiding functional brain organization by identifying unique and divergent properties of overall network organization between the resting-state and task performance. Hum Brain Mapp 38:1992-2007, 2017. © 2017 Wiley Periodicals, Inc.

Disrupted Topological Patterns of Large-Scale Network in Conduct Disorder

Regional abnormalities in brain structure and function, as well as disrupted connectivity, have been found repeatedly in adolescents with conduct disorder (CD). Yet, the large-scale brain topology associated with CD is not well characterized, and little is known about the systematic neural mechanisms of CD. We employed graphic theory to investigate systematically the structural connectivity derived from cortical thickness correlation in a group of patients with CD (N = 43) and healthy controls (HCs, N = 73). Nonparametric permutation tests were applied for between-group comparisons of graphical metrics. Compared with HCs, network measures including global/local efficiency and modularity all pointed to hypo-functioning in CD, despite of preserved small-world organization in both groups. The hubs distribution is only partially overlapped with each other. These results indicate that CD is accompanied by both impaired integration and segregation patterns of brain networks, and the distribution of highly connected neural network 'hubs' is also distinct between groups. Such misconfiguration extends our understanding regarding how structural neural network disruptions may underlie behavioral disturbances in adolescents with CD, and potentially, implicates an aberrant cytoarchitectonic profiles in the brain of CD patients.

Empirical Reference Distributions for Networks of Different Size

Network analysis has become an increasingly prevalent research tool across a vast range of scientific fields. Here, we focus on the particular issue of comparing network statistics, i.e. graph-level measures of network structural features, across multiple networks that differ in size. Although "normalized" versions of some network statistics exist, we demonstrate via simulation why direct comparison is often inappropriate. We consider normalizing network statistics relative to a simple fully parameterized reference distribution and demonstrate via simulation how this is an improvement over direct comparison, but still sometimes problematic. We propose a new adjustment method based on a reference distribution constructed as a mixture model of random graphs which reflect the dependence structure exhibited in the observed networks. We show that using simple Bernoulli models as mixture components in this reference distribution can provide adjusted network statistics that are relatively comparable across different network sizes but still describe interesting features of networks, and that this can be accomplished at relatively low computational expense. Finally, we apply this methodology to a collection of ecological networks derived from the Los Angeles Family and Neighborhood Survey activity location data.

A novel pattern mining approach for identifying cognitive activity in EEG based functional brain networks

The complex nature of neuronal interactions of the human brain has posed many challenges to the research community. To explore the underlying mechanisms of neuronal activity of cohesive brain regions during different cognitive activities, many innovative mathematical and computational models are required. This paper presents a novel Common Functional Pattern Mining approach to demonstrate the similar patterns of interactions due to common behavior of certain brain regions. The electrode sites of EEG-based functional brain network are modeled as a set of transactions and node-based complex network measures as itemsets. These itemsets are transformed into a graph data structure called Functional Pattern Graph. By mining this Functional Pattern Graph, the common functional patterns due to specific brain functioning can be identified. The empirical analyses show the efficiency of the proposed approach in identifying the extent to which the electrode sites (transactions) are similar during various cognitive load states.

Spontaneous physiological variability modulates dynamic functional connectivity in resting-state functional magnetic resonance imaging

It is well known that the blood oxygen level-dependent (BOLD) signal measured by functional magnetic resonance imaging (fMRI) is influenced-in addition to neuronal activity-by fluctuations in physiological signals, including arterial CO2, respiration and heart rate/heart rate variability (HR/HRV). Even spontaneous fluctuations of the aforementioned physiological signals have been shown to influence the BOLD fMRI signal in a regionally specific manner. Related to this, estimates of functional connectivity between different brain regions, performed when the subject is at rest, may be confounded by the effects of physiological signal fluctuations. Moreover, resting functional connectivity has been shown to vary with respect to time (dynamic functional connectivity), with the sources of this variation not fully elucidated. In this context, we examine the relation between dynamic functional connectivity patterns and the time-varying properties of simultaneously recorded physiological signals (end-tidal CO2 and HR/HRV) using resting-state fMRI measurements from 12 healthy subjects. The results reveal a modulatory effect of the aforementioned physiological signals on the dynamic resting functional connectivity patterns for a number of resting-state networks (default mode network, somatosensory, visual). By using discrete wavelet decomposition, we also show that these modulation effects are more pronounced in specific frequency bands.

Brain connectivity in autism spectrum disorder

PURPOSE OF REVIEW

Many studies have reported that individuals with autism spectrum disorder (ASD) have different brain connectivity patterns compared with typically developing individuals. However, the results of more recent studies do not unanimously support the traditional view in which individuals with ASD have lower connectivity between distant brain regions and increased connectivity within local brain regions. In this review, we discuss different methods for measuring brain connectivity and how the use of different metrics may contribute to the lack of convergence of investigations of connectivity in ASD.

RECENT FINDINGS

The discrepancy in brain connectivity results across studies may be due to important methodological factors, such as the connectivity measure applied, the age of patients studied, the brain region(s) examined, and the time interval and frequency band(s) in which connectivity was analyzed.

SUMMARY

We conclude that more sophisticated electroencephalography analytic approaches should be utilized to more accurately infer causation and directionality of information transfer between brain regions, which may show dynamic changes of functional connectivity in the brain. Moreover, further investigations of connectivity with respect to behavior and clinical phenotype are needed to probe underlying brain networks implicated in core deficits of ASD.

Intermediate phenotypes and biomarkers of treatment outcome in major depressive disorder

Major depressive disorder (MDD) is a pleomorphic illness originating from gene x environment interactions. Patients with differing symptom phenotypes receive the same diagnosis and similar treatment recommendations without regard to genomics, brain structure or function, or other physiologic or psychosocial factors. Using this present approach, only one third of patients enter remission with the first medication prescribed, and patients may take longer than 1 year to enter remission with repeated trials. Research to improve treatment effectiveness recently has focused on identification of intermediate phenotypes (IPs) that could parse the heterogeneous population of patients with MDD into subgroups with more homogeneous responses to treatment. Such IPs could be used to develop biomarkers that could be applied clinically to match patients with the treatment that would be most likely to lead to remission. Putative biomarkers include genetic polymorphisms, RNA and protein expression (transcriptome and proteome), neurotransmitter levels (metabolome), additional measures of signaling cascades, oscillatory synchrony, neuronal circuits and neural pathways (connectome), along with other possible physiologic measures. All of these measures represent components of a continuum that extends from proximity to the genome to proximity to the clinical phenotype of depression, and there are many levels along this continuum at which useful IPs may be defined. Because of the highly integrative nature of brain systems and the complex neurobiology of depression, the most useful biomarkers are likely to be those with intermediate proximity both to the genome and the clinical phenotype of MDD. Translation of findings across the spectrum from genotype to phenotype promises to better characterize the complex disruptions in signaling and neuroplasticity that accompany MDD, and ultimately to lead to greater understanding of the causes of depressive illness.

Complex network analysis of resting state EEG in amnestic mild cognitive impairment patients with type 2 diabetes

Purpose: Diabetes is a great risk factor for dementia and mild cognitive impairment (MCI). This study investigates whether complex network-derived features in resting state EEG (rsEEG) could be applied as a biomarker to distinguish amnestic mild cognitive impairment (aMCI) from normal cognitive function in subjects with type 2 diabetes (T2D).

Method: In this study, EEG was recorded in 28 patients with T2D (16 aMCI patients and 12 controls) during a no-task eyes-closed resting state. Pair-wise synchronization of rsEEG signals were assessed in six frequency bands (delta, theta, lower alpha, upper alpha, beta, and gamma) using phase lag index (PLI) and grouped into long distance (intra- and inter-hemispheric) and short distance interactions. PLI-weighted connectivity networks were also constructed, and characterized by mean clustering coefficient and path length. The correlation of these features and Montreal Cognitive Assessment (MoCA) scores was assessed.

Results: Main findings of this study were as follows: (1) In comparison with controls, patients with aMCI had a significant decrease of global mean PLI in lower alpha, upper alpha, and beta bands. Lower functional connection at short and long intra-hemispheric distance mainly appeared on the left hemisphere. (2) In the lower alpha band, clustering coefficient was significantly lower in aMCI group, and the path length significantly increased. (3) Cognitive status measured by MoCA had a significant positive correlation with cluster coefficient and negative correlation with path length in lower alpha band.

Conclusions: The brain network of aMCI patients displayed a disconnection syndrome and a loss of small-world architecture. The correlation between cognitive states and network characteristics suggested that the more in deterioration of the diabetes patients' cognitive state, the less optimal the network organization become. Hence, the complex network-derived biomarkers based on EEG could be employed to track cognitive function of diabetic patients and provide a new diagnosis tool for aMCI.

Whole-brain functional hypoconnectivity as an endophenotype of autism in adolescents

Endophenotypes are heritable and quantifiable markers that may assist in the identification of the complex genetic underpinnings of psychiatric conditions. Here we examined global hypoconnectivity as an endophenotype of autism spectrum conditions (ASCs). We studied well-matched groups of adolescent males with autism, genetically-related siblings of individuals with autism, and typically-developing control participants. We parcellated the brain into 258 regions and used complex-network analysis to detect a robust hypoconnectivity endophenotype in our participant group. We observed that whole-brain functional connectivity was highest in controls, intermediate in siblings, and lowest in ASC, in task and rest conditions. We identified additional, local endophenotype effects in specific networks including the visual processing and default mode networks. Our analyses are the first to show that whole-brain functional hypoconnectivity is an endophenotype of autism in adolescence, and may thus underlie the heritable similarities seen in adolescents with ASC and their relatives.

Heritability of the network architecture of intrinsic brain functional connectivity

The brain's functional network exhibits many features facilitating functional specialization, integration, and robustness to attack. Using graph theory to characterize brain networks, studies demonstrate their small-world, modular, and "rich-club" properties, with deviations reported in many common neuropathological conditions. Here we estimate the heritability of five widely used graph theoretical metrics (mean clustering coefficient (γ), modularity (Q), rich-club coefficient (ϕnorm), global efficiency (λ), small-worldness (σ)) over a range of connection densities (k=5-25%) in a large cohort of twins (N=592, 84 MZ and 89 DZ twin pairs, 246 single twins, age 23 ± 2.5). We also considered the effects of global signal regression (GSR). We found that the graph metrics were moderately influenced by genetic factors h(2) (γ=47-59%, Q=38-59%, ϕnorm=0-29%, λ=52-64%, σ=51-59%) at lower connection densities (≤ 15%), and when global signal regression was implemented, heritability estimates decreased substantially h(2) (γ=0-26%, Q=0-28%, ϕnorm=0%, λ=23-30%, σ=0-27%). Distinct network features were phenotypically correlated (|r|=0.15-0.81), and γ, Q, and λ were found to be influenced by overlapping genetic factors. Our findings suggest that these metrics may be potential endophenotypes for psychiatric disease and suitable for genetic association studies, but that genetic effects must be interpreted with respect to methodological choices.

A Multi-Dimensional and Integrative Approach to Examining the High-Risk and Ultra-High-Risk Stages of Bipolar Disorder

BACKGROUND

Validating the high-risk (HR) and ultra-high-risk (UHR) stages of bipolar disorder (BP) may help enable early intervention strategies.

METHODS

We followed up with 44 offspring of parents with BP, subdividing into the HR and UHR categories. The offspring were aged 8-28 years and were free of any current DSM-IV diagnoses. Our multilevel, integrative approach encompassed gray matter (GM) volumes, brain network connectivity, neuropsychological performance, and clinical outcomes.

FINDINGS

Compared with the healthy controls (HCs) (n = 33), the HR offspring (n = 26) showed GM volume reductions in the right orbitofrontal cortex. Compared with the HR offspring, the UHR offspring (n = 18) exhibited increased GM volumes in four regions. Both the HR and UHR offspring displayed abnormalities in the inferior occipital cortex regarding the measures of degree and centrality, reflecting the connections and roles of the region, respectively. In the UHR versus the HR offspring, the UHR offspring exhibited upwards-shifted small world topologies that reflect high clustering and efficiency in the brain networks. Compared with the HCs, the UHR offspring had significantly lower assortativity, which was suggestive of vulnerability. Finally, processing speed, visual-spatial, and general function were impaired in the UHR offspring but not in the HR offspring.

INTERPRETATION

The abnormalities observed in the HR offspring appear to be inherited, whereas those associated with the UHR offspring represent stage-specific changes predisposing them to developing the disorder.

Genetic influences on brain developmental trajectories on neuroimaging studies: from infancy to young adulthood

Human brain development has been studied intensively with neuroimaging. However, little is known about how genes influence developmental brain trajectories, even though a significant number of genes (about 10,000, or approximately one-third) in the human genome are expressed primarily in the brain and during brain development. Interestingly, in addition to showing differential expression among tissues, many genes are differentially expressed across the ages (e.g., antagonistic pleiotropy). Age-specific gene expression plays an important role in several critical events in brain development, including neuronal cell migration, synaptogenesis and neurotransmitter receptor specificity, as well as in aging and neurodegenerative disorders (e.g., Alzheimer disease or amyotrophic lateral sclerosis). In addition, the majority of psychiatric and mental disorders are polygenic, and many have onsets during childhood and adolescence. In this review, we summarize the major findings from neuroimaging studies that link genetics with brain development, from infancy to young adulthood. Specifically, we focus on the heritability of brain structures across the ages, age-related genetic influences on brain development and sex-specific developmental trajectories.

A probabilistic approach for pediatric epilepsy diagnosis using brain functional connectivity networks

Background

The lives of half a million children in the United States are severely affected due to the alterations in their functional and mental abilities which epilepsy causes. This study aims to introduce a novel decision support system for the diagnosis of pediatric epilepsy based on scalp EEG data in a clinical environment.

Methods

A new time varying approach for constructing functional connectivity networks (FCNs) of 18 subjects (7 subjects from pediatric control (PC) group and 11 subjects from pediatric epilepsy (PE) group) is implemented by moving a window with overlap to split the EEG signals into a total of 445 multi-channel EEG segments (91 for PC and 354 for PE) and finding the hypothetical functional connectivity strengths among EEG channels. FCNs are then mapped into the form of undirected graphs and subjected to extraction of graph theory based features. An unsupervised labeling technique based on Gaussian mixtures model (GMM) is then used to delineate the pediatric epilepsy group from the control group.

Results

The study results show the existence of a statistically significant difference (p < 0.0001) between the mean FCNs of PC and PE groups. The system was able to diagnose pediatric epilepsy subjects with the accuracy of 88.8% with 81.8% sensitivity and 100% specificity purely based on exploration of associations among brain cortical regions and without a priori knowledge of diagnosis.

Conclusions

The current study created the potential of diagnosing epilepsy without need for long EEG recording session and time-consuming visual inspection as conventionally employed.

Time-Varying Network Measures in Resting and Task States Using Graph Theoretical Analysis

Recent studies have shown the importance of graph theory in analyzing characteristic features of functional networks of the human brain. However, many of these explorations have focused on static patterns of a representative graph that describe the relatively long-term brain activity. Therefore, this study established and characterized functional networks based on the synchronization likelihood and graph theory. Quasidynamic graphs were constructed simply by dividing a long-term static graph into a sequence of subgraphs that each had a timescale of 1 s. Irregular changes were then used to investigate differences in human brain networks between resting and math-operation states using magnetoencephalography, which may provide insights into the functional substrates underlying logical reasoning. We found that graph properties could differ from brain frequency rhythms, with a higher frequency indicating a lower small-worldness, while changes in human brain state altered the functional networks into more-centralized and segregated distributions according to the task requirements. Time-varying connectivity maps could provide detailed information about the structure distribution. The frontal theta activity represents the essential foundation and may subsequently interact with high-frequency activity in cognitive processing.

Brain enhancement through cognitive training: a new insight from brain connectome

Owing to the recent advances in neurotechnology and the progress in understanding of brain cognitive functions, improvements of cognitive performance or acceleration of learning process with brain enhancement systems is not out of our reach anymore, on the contrary, it is a tangible target of contemporary research. Although a variety of approaches have been proposed, we will mainly focus on cognitive training interventions, in which learners repeatedly perform cognitive tasks to improve their cognitive abilities. In this review article, we propose that the learning process during the cognitive training can be facilitated by an assistive system monitoring cognitive workloads using electroencephalography (EEG) biomarkers, and the brain connectome approach can provide additional valuable biomarkers for facilitating leaners' learning processes. For the purpose, we will introduce studies on the cognitive training interventions, EEG biomarkers for cognitive workload, and human brain connectome. As cognitive overload and mental fatigue would reduce or even eliminate gains of cognitive training interventions, a real-time monitoring of cognitive workload can facilitate the learning process by flexibly adjusting difficulty levels of the training task. Moreover, cognitive training interventions should have effects on brain sub-networks, not on a single brain region, and graph theoretical network metrics quantifying topological architecture of the brain network can differentiate with respect to individual cognitive states as well as to different individuals' cognitive abilities, suggesting that the connectome is a valuable approach for tracking the learning progress. Although only a few studies have exploited the connectome approach for studying alterations of the brain network induced by cognitive training interventions so far, we believe that it would be a useful technique for capturing improvements of cognitive functions.