Bayesian inference of structural brain networks

Duality between predictability and reconstructability in complex systems

Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. We provide analytical calculations of the uncertainty coefficients for many different systems, including continuous deterministic systems, and describe a numerical procedure when exact calculations are intractable. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process. Finally, we provide evidence that predictability-reconstruction dualities may exist in dynamical processes on real networks close to criticality.

Constraining functional coactivation with a cluster-based structural connectivity network

In this article, we propose a two-step pipeline to explore task-dependent functional coactivations of brain clusters with constraints from the structural connectivity network. In the first step, the pipeline employs a nonparametric Bayesian clustering method that can estimate the optimal number of clusters, cluster assignments of brain regions of interest (ROIs), and the strength of within- and between-cluster connections without any prior knowledge. In the second step, a factor analysis model is applied to functional data with factors defined as the obtained structural clusters and the factor structure informed by the structural network. The coactivations of ROIs and their clusters can be studied by correlations between factors, which can largely differ by ongoing cognitive task. We provide a simulation study to validate that the pipeline can recover the underlying structural and functional network. We also apply the proposed pipeline to empirical data to explore the structural network of ROIs obtained by the Gordon parcellation and study their functional coactivations across eight cognitive tasks and a resting-state condition.

Identification of Pattern Completion Neurons in Neuronal Ensembles Using Probabilistic Graphical Models

Neuronal ensembles are groups of neurons with coordinated activity that could represent sensory, motor, or cognitive states. The study of how neuronal ensembles are built, recalled, and involved in the guiding of complex behaviors has been limited by the lack of experimental and analytical tools to reliably identify and manipulate neurons that have the ability to activate entire ensembles. Such pattern completion neurons have also been proposed as key elements of artificial and biological neural networks. Indeed, the relevance of pattern completion neurons is highlighted by growing evidence that targeting them can activate neuronal ensembles and trigger behavior. As a method to reliably detect pattern completion neurons, we use conditional random fields (CRFs), a type of probabilistic graphical model. We apply CRFs to identify pattern completion neurons in ensembles in experiments using in vivo two-photon calcium imaging from primary visual cortex of male mice and confirm the CRFs predictions with two-photon optogenetics. To test the broader applicability of CRFs we also analyze publicly available calcium imaging data (Allen Institute Brain Observatory dataset) and demonstrate that CRFs can reliably identify neurons that predict specific features of visual stimuli. Finally, to explore the scalability of CRFs we apply them to in silico network simulations and show that CRFs-identified pattern completion neurons have increased functional connectivity. These results demonstrate the potential of CRFs to characterize and selectively manipulate neural circuits.SIGNIFICANCE STATEMENT We describe a graph theory method to identify and optically manipulate neurons with pattern completion capability in mouse cortical circuits. Using calcium imaging and two-photon optogenetics in vivo we confirm that key neurons identified by this method can recall entire neuronal ensembles. This method could be broadly applied to manipulate neuronal ensemble activity to trigger behavior or for therapeutic applications in brain prostheses.

Grant Report on the Effects of Childhood Maltreatment on Neurocircuitry in Adolescent Depression

This report describes the protocol for an ongoing project funded by the National Institutes of Health (R01MH108155) that is focused on effects of childhood maltreatment (MALTX) on neurocircuitry changes associated with adolescent major depressive disorder (MDD). Extant clinical and neuroimaging literature on MDD is reviewed, which has relied on heterogeneous samples that do not parse out the unique contribution of MALTX on neurobiological changes in MDD. Employing a 2 × 2 study design (controls with no MALTX or MDD, MALTX only, MDD only, and MDD + MALTX), and based on a cohesive theoretical model that incorporates behavioral, cognitive and neurobiological domains, we describe the multi-modal neuroimaging techniques used to test whether structural and functional alterations in the fronto-limbic and fronto-striatal circuits associated with adolescent MDD are moderated by MALTX. We hypothesize that MDD + MALTX youth will show alterations in the fronto-limbic circuit, with reduced connectivity between the amygdala (AMG) and the prefrontal cortex (PFC), as the AMG is sensitive to stress/threat during development. Participants with MDD will exhibit increased functional connectivity between the AMG and PFC due to self-referential negative emotions. Lastly, MDD + MALTX will only show changes in motivational/anticipatory aspects of the fronto-striatal circuit, and MDD will exhibit changes in motivational and consummatory/outcome aspects of reward-processing. Our goal is to identify distinct neural substrates associated with MDD due to MALTX compared to other causes, as these markers could be used to more effectively predict treatment outcome, index treatment response, and facilitate alternative treatments for adolescents who do not respond well to traditional approaches.

Relative rate of change in cognitive score network dynamics via Bayesian hierarchical models reveal spatial patterns of neurodegeneration

The degeneration of the human brain is a complex process, which often affects certain brain regions due to healthy aging or disease. This degeneration can be evaluated on regions of interest (ROI) in the brain through probabilistic networks and morphological estimates. Current approaches for finding such networks are limited to analyses at discrete neuropsychological stages, which cannot appropriately account for connectivity dynamics over the onset of cognitive deterioration, and morphological changes are seldom unified with connectivity networks, despite known dependencies. To overcome these limitations, a probabilistic wombling model is proposed to simultaneously estimate ROI cortical thickness and covariance networks contingent on rates of change in cognitive decline. This proposed model was applied to analyze longitudinal data from healthy control (HC) and Alzheimer's disease (AD) groups and found connection differences pertaining to regions, which play a crucial role in lasting cognitive impairment, such as the entorhinal area and temporal regions. Moreover, HC cortical thickness estimates were significantly higher than those in the AD group across all ROIs. The analyses presented in this work will help practitioners jointly analyze brain tissue atrophy at the ROI-level conditional on neuropsychological networks, which could potentially allow for more targeted therapeutic interventions.

Dynamic spatiotemporal patterns of brain connectivity reorganize across development

Late human development is characterized by the maturation of high-level functional processes, which rely on reshaping of white matter connections, as well as synaptic density. However, the relationship between the whole-brain dynamics and the underlying white matter networks in neurodevelopment is largely unknown. In this study, we focused on how the structural connectome shapes the emerging dynamics of cerebral development between the ages of 6 and 33 years, using functional and diffusion magnetic resonance imaging combined into a spatiotemporal connectivity framework. We defined two new measures of brain dynamics, namely the system diversity and the spatiotemporal diversity, which quantify the level of integration/segregation between functional systems and the level of temporal self-similarity of the functional patterns of brain dynamics, respectively. We observed a global increase in system diversity and a global decrease and local refinement in spatiotemporal diversity values with age. In support of these findings, we further found an increase in the usage of long-range and inter-system white matter connectivity and a decrease in the usage of short-range connectivity with age. These findings suggest that dynamic functional patterns in the brain progressively become more integrative and temporally self-similar with age. These functional changes are supported by a greater involvement of long-range and inter-system axonal pathways.

Structural and Functional Connectivity Underlying Gray Matter Covariance: Impact of Developmental Insult

Structural covariance networks (SCNs) may offer unique insights into the developmental impact of childhood maltreatment (CM) because they are thought to reflect coordinated maturation of distinct gray matter regions. T1-weighted magnetic resonance images were acquired from 121 young people with emerging mental illness. Diffusion-weighted and resting-state functional imaging was also acquired from a random subset of participants (n = 62). Ten study-specific SCNs were identified using a whole-brain gray matter independent component analysis. The effects of CM and age on average gray matter density and the expression of each SCN were calculated. CM was linked to age-related decreases in gray matter density across an SCN that overlapped with the default mode network (DMN) and frontoparietal network. Resting-state functional connectivity (rsFC) and structural connectivity were calculated in the study-specific SCN and across the whole brain. Gray matter covariance was significantly correlated with rsFC across the SCN, and rsFC fully mediated the relationship between gray matter covariance and structural connectivity in the nonmaltreated group. A unique association of gray matter covariance with structural connectivity was detected among individuals with a history of CM. Perturbation of gray matter development across the DMN and frontoparietal network following CM may have significant implications for mental well-being, given the networks' roles in self-referential activity. Cross-modal comparisons suggest that reduced gray matter following CM could arise from deficient functional activity earlier in life.

Topological structures are consistently overestimated in functional complex networks

Functional complex networks have meant a pivotal change in the way we understand complex systems, being the most outstanding one the human brain. These networks have classically been reconstructed using a frequentist approach that, while simple, completely disregards the uncertainty that derives from data finiteness. We provide here an alternative solution based on Bayesian inference, with link weights treated as random variables described by probability distributions, from which ensembles of networks are sampled. By using both statistical and topological considerations, we prove that the role played by links’ uncertainty is equivalent to the introduction of a random rewiring, whose omission leads to a consistent overestimation of topological structures. We further show that this bias is enhanced in short time series, suggesting the existence of a theoretical time resolution limit for obtaining reliable structures. We also propose a simple sampling process for correcting topological values obtained in frequentist networks. We finally validate these concepts through synthetic and real network examples, the latter representing the brain electrical activity of a group of people during a cognitive task.

Multi-subject hierarchical inverse covariance modelling improves estimation of functional brain networks

A Bayesian model for sparse, hierarchical, inver-covariance estimation is presented, and applied to multi-subject functional connectivity estimation in the human brain. It enables simultaneous inference of the strength of connectivity between brain regions at both subject and population level, and is applicable to fMRI, MEG and EEG data. Two versions of the model can encourage sparse connectivity, either using continuous priors to suppress irrelevant connections, or using an explicit description of the network structure to estimate the connection probability between each pair of regions. A large evaluation of this model, and thirteen methods that represent the state of the art of inverse covariance modelling, is conducted using both simulated and resting-state functional imaging datasets. Our novel Bayesian approach has similar performance to the best extant alternative, Ng et al.'s Sparse Group Gaussian Graphical Model algorithm, which also is based on a hierarchical structure. Using data from the Human Connectome Project, we show that these hierarchical models are able to reduce the measurement error in MEG beta-band functional networks by 10%, producing concomitant increases in estimates of the genetic influence on functional connectivity.

Transient networks of spatio-temporal connectivity map communication pathways in brain functional systems

The study of brain dynamics enables us to characterize the time-varying functional connectivity among distinct neural groups. However, current methods suffer from the absence of structural connectivity information. We propose to integrate infra-slow neural oscillations and anatomical-connectivity maps, as derived from functional and diffusion MRI, in a multilayer-graph framework that captures transient networks of spatio-temporal connectivity. These networks group anatomically wired and temporary synchronized brain regions and encode the propagation of functional activity on the structural connectome. In a group of 71 healthy subjects, we find that these transient networks demonstrate power-law spatial and temporal size, globally organize into well-known functional systems and describe wave-like trajectories of activation across anatomically connected regions. Within the transient networks, activity propagates through polysynaptic paths that include selective ensembles of structural connections and differ from the structural shortest paths. In the light of the communication-through-coherence principle, the identified spatio-temporal networks could encode communication channels' selection and neural assemblies, which deserves further attention. This work contributes to the understanding of brain structure-function relationships by considering the time-varying nature of resting-state interactions on the axonal scaffold, and it offers a convenient framework to study large-scale communication mechanisms and functional dynamics.

Network neuroscience

Despite substantial recent progress, our understanding of the principles and mechanisms underlying complex brain function and cognition remains incomplete. Network neuroscience proposes to tackle these enduring challenges. Approaching brain structure and function from an explicitly integrative perspective, network neuroscience pursues new ways to map, record, analyze and model the elements and interactions of neurobiological systems. Two parallel trends drive the approach: the availability of new empirical tools to create comprehensive maps and record dynamic patterns among molecules, neurons, brain areas and social systems; and the theoretical framework and computational tools of modern network science. The convergence of empirical and computational advances opens new frontiers of scientific inquiry, including network dynamics, manipulation and control of brain networks, and integration of network processes across spatiotemporal domains. We review emerging trends in network neuroscience and attempt to chart a path toward a better understanding of the brain as a multiscale networked system.

The missing link: Predicting connectomes from noisy and partially observed tract tracing data

Our understanding of the wiring map of the brain, known as the connectome, has increased greatly in the last decade, mostly due to technological advancements in neuroimaging techniques and improvements in computational tools to interpret the vast amount of available data. Despite this, with the exception of the C. elegans roundworm, no definitive connectome has been established for any species. In order to obtain this, tracer studies are particularly appealing, as these have proven highly reliable. The downside of tract tracing is that it is costly to perform, and can only be applied ex vivo. In this paper, we suggest that instead of probing all possible connections, hitherto unknown connections may be predicted from the data that is already available. Our approach uses a 'latent space model' that embeds the connectivity in an abstract physical space. Regions that are close in the latent space have a high chance of being connected, while regions far apart are most likely disconnected in the connectome. After learning the latent embedding from the connections that we did observe, the latent space allows us to predict connections that have not been probed previously. We apply the methodology to two connectivity data sets of the macaque, where we demonstrate that the latent space model is successful in predicting unobserved connectivity, outperforming two baselines and an alternative model in nearly all cases. Furthermore, we show how the latent spatial embedding may be used to integrate multimodal observations (i.e. anterograde and retrograde tracers) for the mouse neocortex. Finally, our probabilistic approach enables us to make explicit which connections are easy to predict and which prove difficult, allowing for informed follow-up studies.

Unifying Inference of Meso-Scale Structures in Networks

Networks are among the most prevalent formal representations in scientific studies, employed to depict interactions between objects such as molecules, neuronal clusters, or social groups. Studies performed at meso-scale that involve grouping of objects based on their distinctive interaction patterns form one of the main lines of investigation in network science. In a social network, for instance, meso-scale structures can correspond to isolated social groupings or groups of individuals that serve as a communication core. Currently, the research on different meso-scale structures such as community and core-periphery structures has been conducted via independent approaches, which precludes the possibility of an algorithmic design that can handle multiple meso-scale structures and deciding which structure explains the observed data better. In this study, we propose a unified formulation for the algorithmic detection and analysis of different meso-scale structures. This facilitates the investigation of hybrid structures that capture the interplay between multiple meso-scale structures and statistical comparison of competing structures, all of which have been hitherto unavailable. We demonstrate the applicability of the methodology in analyzing the human brain network, by determining the dominant organizational structure (communities) of the brain, as well as its auxiliary characteristics (core-periphery).

Bayesian Estimation of Conditional Independence Graphs Improves Functional Connectivity Estimates

Functional connectivity concerns the correlated activity between neuronal populations in spatially segregated regions of the brain, which may be studied using functional magnetic resonance imaging (fMRI). This coupled activity is conveniently expressed using covariance, but this measure fails to distinguish between direct and indirect effects. A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections. Importantly, provided the data are normally distributed, if two variables are conditionally independent given all other variables, their respective partial correlation is zero. In this paper, we propose a probabilistic generative model that allows us to estimate functional connectivity in terms of both partial correlations and a graph representing conditional independencies. Simulation results show that this methodology is able to outperform the graphical LASSO, which is the de facto standard for estimating partial correlations. Furthermore, we apply the model to estimate functional connectivity for twenty subjects using resting-state fMRI data. Results show that our model provides a richer representation of functional connectivity as compared to considering partial correlations alone. Finally, we demonstrate how our approach can be extended in several ways, for instance to achieve data fusion by informing the conditional independence graph with data from probabilistic tractography. As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results. This reveals that while we are able to infer a clear backbone of connectivity in our empirical results, the data are not accurately described by simply looking at the mode of the distribution over connectivity. The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

Significant neuroscientific effort is devoted to elucidating functional connectivity between spatially segregated brain regions. This requires that we are able to quantify the degree of dependence between the signals of different areas. Yet how this must be accomplished—using which measures, each with their own limitations and interpretations—is far from a trivial task. One frequently advocated metric for functional connectivity is partial correlation, which is related to conditional independence: if two regions are independent, conditioned on all other regions, then their partial correlation is zero, assuming Gaussian data. Here, we use a probabilistic generative model to describe the relationship between functional connectivity and conditional independence. We apply this Bayesian approach to reveal functional connectivity between subcortical areas, and in addition we propose different variants of the generative model for connectivity. In the first, we address how a Bayesian formulation of connectivity allows for integration with other imaging modalities, resulting in data fusion. Secondly, we show how prior constraints can be incorporated in our estimates of connectivity.

Probabilistic clustering of the human connectome identifies communities and hubs

A fundamental assumption in neuroscience is that brain function is constrained by its structural properties. This motivates the idea that the brain can be parcellated into functionally coherent regions based on anatomical connectivity patterns that capture how different areas are interconnected. Several studies have successfully implemented this idea in humans using diffusion weighted MRI, allowing parcellation to be conducted in vivo. Two distinct approaches to connectivity-based parcellation can be identified. The first uses the connection profiles of brain regions as a feature vector, and groups brain regions with similar connection profiles together. Alternatively, one may adopt a network perspective that aims to identify clusters of brain regions that show dense within-cluster and sparse between-cluster connectivity. In this paper, we introduce a probabilistic model for connectivity-based parcellation that unifies both approaches. Using the model we are able to obtain a parcellation of the human brain whose clusters may adhere to either interpretation. We find that parts of the connectome consistently cluster as densely connected components, while other parts consistently result in clusters with similar connections. Interestingly, the densely connected components consist predominantly of major cortical areas, while the clusters with similar connection profiles consist of regions that have previously been identified as the 'rich club'; regions known for their integrative role in connectivity. Furthermore, the probabilistic model allows quantification of the uncertainty in cluster assignments. We show that, while most clusters are clearly delineated, some regions are more difficult to assign. These results indicate that care should be taken when interpreting connectivity-based parcellations obtained using alternative deterministic procedures.

Relating structural and functional connectivity in MRI: a simple model for a complex brain

Advances in magnetic resonance imaging (MRI) allow to gain critical insight into the structure of neural networks and their functional dynamics. To relate structural connectivity [as quantified by diffusion-weighted imaging (DWI) tractography] and functional connectivity [as obtained from functional MRI (fMRI)], increasing emphasis has been put on computational models of brain activity. In the present study, we use structural equation modeling (SEM) with structural connectivity to predict functional connectivity. The resulting model takes the simple form of a spatial simultaneous autoregressive model (sSAR), whose parameters can be estimated in a Bayesian framework. On synthetic data, results showed very good accuracy and reliability of the inference process. On real data, we found that the sSAR performed significantly better than two other reference models as well as than structural connectivity alone, but that the Bayesian procedure did not bring significant improvement in fit compared to two simpler approaches. Nonetheless, we also found that the values of the region-specific parameters inferred using Bayesian inference differed significantly across resting-state networks. These results demonstrate 1) that a simple abstract model is able to perform better that more complex models based on more realistic assumptions, 2) that the parameters of the sSAR can be estimated and can potentially be used as biomarkers, but also 3) that the sSAR, while being the best-performing model, is at best still a very crude model of the relationship between structure and function in MRI.

Quantifying uncertainty in brain network measures using Bayesian connectomics

The wiring diagram of the human brain can be described in terms of graph measures that characterize structural regularities. These measures require an estimate of whole-brain structural connectivity for which one may resort to deterministic or thresholded probabilistic streamlining procedures. While these procedures have provided important insights about the characteristics of human brain networks, they ultimately rely on unwarranted assumptions such as those of noise-free data or the use of an arbitrary threshold. Therefore, resulting structural connectivity estimates as well as derived graph measures fail to fully take into account the inherent uncertainty in the structural estimate. In this paper, we illustrate an easy way of obtaining posterior distributions over graph metrics using Bayesian inference. It is shown that this posterior distribution can be used to quantify uncertainty about graph-theoretical measures at the single subject level, thereby providing a more nuanced view of the graph-theoretical properties of human brain connectivity. We refer to this model-based approach to connectivity analysis as Bayesian connectomics.

Identification of optimal structural connectivity using functional connectivity and neural modeling

The complex network dynamics that arise from the interaction of the brain's structural and functional architectures give rise to mental function. Theoretical models demonstrate that the structure-function relation is maximal when the global network dynamics operate at a critical point of state transition. In the present work, we used a dynamic mean-field neural model to fit empirical structural connectivity (SC) and functional connectivity (FC) data acquired in humans and macaques and developed a new iterative-fitting algorithm to optimize the SC matrix based on the FC matrix. A dramatic improvement of the fitting of the matrices was obtained with the addition of a small number of anatomical links, particularly cross-hemispheric connections, and reweighting of existing connections. We suggest that the notion of a critical working point, where the structure-function interplay is maximal, may provide a new way to link behavior and cognition, and a new perspective to understand recovery of function in clinical conditions.

Anatomical brain networks on the prediction of abnormal brain states

Graph-based brain anatomical network analysis models the brain as a graph whose nodes represent structural/functional regions, whereas the links between them represent nervous fiber connections. Initial studies of brain anatomical networks using this approach were devoted to describe the key organizational principles of the normal brain, while current trends seem to be more focused on detecting network alterations associated to specific brain disorders. Anatomical networks reconstructed using diffusion-weighed magnetic resonance-imaging techniques can be particularly useful in predicting abnormal brain states in which the white matter structure and, subsequently, the interconnections between gray matter regions are altered (e.g., due to the presence of diseases such as schizophrenia, stroke, multiple sclerosis, and dementia). This article offers an overview from early gross connectional anatomy explorations until more recent advances on anatomical brain network reconstruction approaches, with a specific focus on how the latter move toward the prediction of abnormal brain states. While anatomical graph-based predictor approaches are still at an early stage, they bear promising implications for individualized clinical diagnosis of neurological and psychiatric disorders, as well as for neurodevelopmental evaluations and subsequent assisted creation of educational strategies related to specific cognitive disorders.