On how network architecture determines the dominant patterns of spontaneous neural activity

A multi-modal, asymmetric, weighted, and signed description of anatomical connectivity

The macroscale connectome is the network of physical, white-matter tracts between brain areas. The connections are generally weighted and their values interpreted as measures of communication efficacy. In most applications, weights are either assigned based on imaging features-e.g. diffusion parameters-or inferred using statistical models. In reality, the ground-truth weights are unknown, motivating the exploration of alternative edge weighting schemes. Here, we explore a multi-modal, regression-based model that endows reconstructed fiber tracts with directed and signed weights. We find that the model fits observed data well, outperforming a suite of null models. The estimated weights are subject-specific and highly reliable, even when fit using relatively few training samples, and the networks maintain a number of desirable features. In summary, we offer a simple framework for weighting connectome data, demonstrating both its ease of implementation while benchmarking its utility for typical connectome analyses, including graph theoretic modeling and brain-behavior associations.

Functional Connectivity Prediction With Deep Learning for Graph Transformation

Inferring resting-state functional connectivity (FC) from anatomical brain wiring, known as structural connectivity (SC), is of enormous significance in neuroscience for understanding biological neuronal networks and treating mental diseases. Both SC and FC are networks where the nodes are brain regions, and in SC, the edges are the physical fiber nerves among the nodes, while in FC, the edges are the nodes' coactivation relations. Despite the importance of SC and FC, until very recently, the rapidly growing research body on this topic has generally focused on either linear models or computational models that rely heavily on heuristics and simple assumptions regarding the mapping between FC and SC. However, the relationship between FC and SC is actually highly nonlinear and complex and contains considerable randomness; additional factors, such as the subject's age and health, can also significantly impact the SC-FC relationship and hence cannot be ignored. To address these challenges, here, we develop a novel SC-to-FC generative adversarial network (SF-GAN) framework for mapping SC to FC, along with additional metafeatures based on a newly proposed graph neural network-based generative model that is capable of learning the stochasticity. Specifically, a new graph-based conditional generative adversarial nets model is proposed, where edge convolution layers are leveraged to encode the graph patterns in the SC in the form of a graph representation. New edge deconvolution layers are then utilized to decode the representation back to FC. Additional metafeatures of subjects' profile information are integrated into the graph representation with newly designed sparse-regularized layers that can automatically select features that impact FC. Finally, we have also proposed new post hoc explainer of our SF-GAN, which can identify which subgraphs in SC strongly influence which subgraphs in FC by a new multilevel edge-correlation-guided graph clustering problem. The results of experiments conducted to test the new model confirm that it significantly outperforms existing state-of-the-art methods, with additional interpretability for identifying important metafeatures and subgraphs.

The impact of input node placement in the controllability of structural brain networks

Network controllability refers to the ability to steer the state of a network towards a target state by driving certain nodes, known as input nodes. This concept can be applied to brain networks for studying brain function and its relation to the structure, which has numerous practical applications. Brain network controllability involves using external signals such as electrical stimulation to drive specific brain regions and navigate the neurophysiological activity level of the brain around the state space. Although controllability is mainly theoretical, the energy required for control is critical in real-world implementations. With a focus on the structural brain networks, this study explores the impact of white matter fiber architecture on the control energy in brain networks using the theory of how input node placement affects the LCC (the longest distance between inputs and other network nodes). Initially, we use a single input node as it is theoretically possible to control brain networks with just one input. We show that highly connected brain regions that lead to lower LCCs are more energy-efficient as a single input node. However, there may still be a need for a significant amount of control energy with one input, and achieving controllability with less energy could be of interest. We identify the minimum number of input nodes required to control brain networks with smaller LCCs, demonstrating that reducing the LCC can significantly decrease the control energy in brain networks. Our results show that relying solely on highly connected nodes is not effective in controlling brain networks with lower energy by using multiple inputs because of densely interconnected brain network hubs. Instead, a combination of low and high-degree nodes is necessary.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Analytic relationship of relative synchronizability to network structure and motifs

Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein-Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks.

Uncovering the biological basis of control energy: Structural and metabolic correlates of energy inefficiency in temporal lobe epilepsy

Network control theory is increasingly used to profile the brain's energy landscape via simulations of neural dynamics. This approach estimates the control energy required to simulate the activation of brain circuits based on structural connectome measured using diffusion magnetic resonance imaging, thereby quantifying those circuits' energetic efficiency. The biological basis of control energy, however, remains unknown, hampering its further application. To fill this gap, investigating temporal lobe epilepsy as a lesion model, we show that patients require higher control energy to activate the limbic network than healthy volunteers, especially ipsilateral to the seizure focus. The energetic imbalance between ipsilateral and contralateral temporolimbic regions is tracked by asymmetric patterns of glucose metabolism measured using positron emission tomography, which, in turn, may be selectively explained by asymmetric gray matter loss as evidenced in the hippocampus. Our investigation provides the first theoretical framework unifying gray matter integrity, metabolism, and energetic generation of neural dynamics.

A perturbative approach to study information communication in brain networks

How communication among neuronal ensembles shapes functional brain dynamics is a question of fundamental importance to neuroscience. Communication in the brain can be viewed as a product of the interaction of node activities with the structural network over which these activities flow. The study of these interactions is, however, restricted by the difficulties in describing the complex dynamics of the brain. There is thus a need to develop methods to study these network-dynamical interactions and how they impact information flow, without having to ascertain dynamics a priori or resort to restrictive analytical approaches. Here, we adapt a recently established network analysis method based on perturbations, it to a neuroscientific setting to study how information flow in the brain can raise from properties of underlying structure. For proof-of-concept, we apply the approach on in silico whole-brain models. We expound on the functional implications of the distributions of metrics that capture network-dynamical interactions, termed net influence and flow. We also study the network-dynamical interactions at the level of resting-state networks. An attractive feature of this method is its simplicity, which allows a direct translation to an experimental or clinical setting, such as for identifying targets for stimulation studies or therapeutic interventions.

Reliability and subject specificity of personalized whole-brain dynamical models

Dynamical whole-brain models were developed to link structural (SC) and functional connectivity (FC) together into one framework. Nowadays, they are used to investigate the dynamical regimes of the brain and how these relate to behavioral, clinical and demographic traits. However, there is no comprehensive investigation on how reliable and subject specific the modeling results are given the variability of the empirical FC. In this study, we show that the parameters of these models can be fitted with a "poor" to "good" reliability depending on the exact implementation of the modeling paradigm. We find, as a general rule of thumb, that enhanced model personalization leads to increasingly reliable model parameters. In addition, we observe no clear effect of the model complexity evaluated by separately sampling results for linear, phase oscillator and neural mass network models. In fact, the most complex neural mass model often yields modeling results with "poor" reliability comparable to the simple linear model, but demonstrates an enhanced subject specificity of the model similarity maps. Subsequently, we show that the FC simulated by these models can outperform the empirical FC in terms of both reliability and subject specificity. For the structure-function relationship, simulated FC of individual subjects may be identified from the correlations with the empirical SC with an accuracy up to 70%, but not vice versa for non-linear models. We sample all our findings for 8 distinct brain parcellations and 6 modeling conditions and show that the parcellation-induced effect is much more pronounced for the modeling results than for the empirical data. In sum, this study provides an exploratory account on the reliability and subject specificity of dynamical whole-brain models and may be relevant for their further development and application. In particular, our findings suggest that the application of the dynamical whole-brain modeling should be tightly connected with an estimate of the reliability of the results.

Integrals and series related to propagators of neural and haemodynamic waves

The propagator, or Green function, of a class of neural activity fields and of haemodynamic waves is evaluated exactly. The results enable a number of related integrals to be evaluated, along with series expansions of key results in terms of Bessel functions of the second kind. Connections to other related equations are also noted.

Discrete spectral eigenmode-resonance network of brain dynamics and connectivity

The problem of finding a compact natural representation of brain dynamics and connectivity is addressed using an expansion in terms of physical spatial eigenmodes and their frequency resonances. It is demonstrated that this discrete expansion via the system transfer function enables linear and nonlinear dynamics to be analyzed in compact form in terms of natural dynamic "atoms," each of which is a frequency resonance of an eigenmode. Because these modal resonances are determined by the system dynamics, not the investigator, they are privileged over widely used phenomenological patterns, and obviate the need for artificial discretizations and thresholding in coordinate space. It is shown that modal resonances participate as nodes of a discrete spectral network, are noninteracting in the linear regime, but are linked nonlinearly by wave-wave coalescence and decay processes. The modal resonance formulation is shown to be capable of speeding numerical calculations of strongly nonlinear interactions. Recent work in brain dynamics, especially based on neural field theory (NFT) approaches, allows eigenmodes and their resonances to be estimated from data without assuming a specific brain model. This means that dynamic equations can be inferred using system identification methods from control theory, rather than being assumed, and resonances can be interpreted as control-systems data filters. The results link brain activity and connectivity with control-systems functions such as prediction and attention via gain control and can also be linked to specific NFT predictions if desired, thereby providing a convenient bridge between physiologically based theories and experiment. Amplitudes of modes and resonances can also be tracked to provide a more direct and temporally localized representation of the dynamics than correlations and covariances, which are widely used in the field. By synthesizing many different lines of research, this work provides a way to link quantitative electrophysiological and imaging measurements, connectivity, brain dynamics, and function. This underlines the need to move between coordinate and spectral representations as required. Moreover, standard theoretical-physics approaches and mathematical methods can be used in place of ad hoc statistical measures such as those based on graph theory of artificially discretized and decimated networks, which are highly prone to selection effects and artifacts.

Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators

We apply the framework of optimal nonlinear control to steer the dynamics of a whole-brain network of FitzHugh-Nagumo oscillators. Its nodes correspond to the cortical areas of an atlas-based segmentation of the human cerebral cortex, and the internode coupling strengths are derived from diffusion tensor imaging data of the connectome of the human brain. Nodes are coupled using an additive scheme without delays and are driven by background inputs with fixed mean and additive Gaussian noise. Optimal control inputs to nodes are determined by minimizing a cost functional that penalizes the deviations from a desired network dynamic, the control energy, and spatially nonsparse control inputs. Using the strength of the background input and the overall coupling strength as order parameters, the network's state-space decomposes into regions of low- and high-activity fixed points separated by a high-amplitude limit cycle, all of which qualitatively correspond to the states of an isolated network node. Along the borders, however, additional limit cycles, asynchronous states, and multistability can be observed. Optimal control is applied to several state-switching and network synchronization tasks, and the results are compared to controllability measures from linear control theory for the same connectome. We find that intuitions from the latter about the roles of nodes in steering the network dynamics, which are solely based on connectome features, do not generally carry over to nonlinear systems, as had been previously implied. Instead, the role of nodes under optimal nonlinear control critically depends on the specified task and the system's location in state space. Our results shed new light on the controllability of brain network states and may serve as an inspiration for the design of new paradigms for noninvasive brain stimulation.

Principles and open questions in functional brain network reconstruction

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network representation involves often covert theoretical assumptions and methodological choices which affect the way networks are reconstructed from experimental data, and ultimately the resulting network properties and their interpretation. Here, we review some fundamental conceptual underpinnings and technical issues associated with brain network reconstruction, and discuss how their mutual influence concurs in clarifying the organization of brain function.

Determination of Dynamic Brain Connectivity via Spectral Analysis

Spectral analysis based on neural field theory is used to analyze dynamic connectivity via methods based on the physical eigenmodes that are the building blocks of brain dynamics. These approaches integrate over space instead of averaging over time and thereby greatly reduce or remove the temporal averaging effects, windowing artifacts, and noise at fine spatial scales that have bedeviled the analysis of dynamical functional connectivity (FC). The dependences of FC on dynamics at various timescales, and on windowing, are clarified and the results are demonstrated on simple test cases, demonstrating how modes provide directly interpretable insights that can be related to brain structure and function. It is shown that FC is dynamic even when the brain structure and effective connectivity are fixed, and that the observed patterns of FC are dominated by relatively few eigenmodes. Common artifacts introduced by statistical analyses that do not incorporate the physical nature of the brain are discussed and it is shown that these are avoided by spectral analysis using eigenmodes. Unlike most published artificially discretized “resting state networks” and other statistically-derived patterns, eigenmodes overlap, with every mode extending across the whole brain and every region participating in every mode—just like the vibrations that give rise to notes of a musical instrument. Despite this, modes are independent and do not interact in the linear limit. It is argued that for many purposes the intrinsic limitations of covariance-based FC instead favor the alternative of tracking eigenmode coefficients vs. time, which provide a compact representation that is directly related to biophysical brain dynamics.

Structure-informed functional connectivity driven by identifiable and state-specific control regions

Describing how the brain anatomical wiring contributes to the emergence of coordinated neural activity underlying complex behavior remains challenging. Indeed, patterns of remote coactivations that adjust with the ongoing task-demand do not systematically match direct, static anatomical links. Here, we propose that observed coactivation patterns, known as functional connectivity (FC), can be explained by a controllable linear diffusion dynamics defined on the brain architecture. Our model, termed structure-informed FC, is based on the hypothesis that different sets of brain regions controlling the information flow on the anatomical wiring produce state-specific functional patterns. We thus introduce a principled framework for the identification of potential control centers in the brain. We find that well-defined, sparse, and robust sets of control regions, partially overlapping across several tasks and resting state, produce FC patterns comparable to empirical ones. Our findings suggest that controllability is a fundamental feature allowing the brain to reach different states.

Understanding how brain anatomy promotes particular patterns of coactivations among neural regions is a key challenge in neuroscience. This challenge can be addressed using network science and systems theory. Here, we propose that coactivations result from the diffusion of information through the network of anatomical links connecting brain regions, with certain regions controlling the dynamics. We translate this hypothesis into a model called structure-informed functional connectivity, and we introduce a framework for identifying control regions based on empirical data. We find that our model produces coactivation patterns comparable to empirical ones, and that distinct sets of control regions are associated with different functional states. These findings suggest that controllability is an important feature allowing the brain to reach different states.

Neural field theory of neural avalanche exponents

The power-law exponents of observed size and lifetime distributions of near-critical neural avalanches are calculated from neural field theory using diagrammatic methods. This brings neural avalanches within the ambit of neural field theory, which has also previously explained near-critical 1/f spectra and many other observed features of neural activity. This strengthens the case for near-criticality of the brain and opens the way for these other phenomena to be interrelated with avalanches and their dynamics.

Computational Models in Electroencephalography

Computational models lie at the intersection of basic neuroscience and healthcare applications because they allow researchers to test hypotheses in silico and predict the outcome of experiments and interactions that are very hard to test in reality. Yet, what is meant by “computational model” is understood in many different ways by researchers in different fields of neuroscience and psychology, hindering communication and collaboration. In this review, we point out the state of the art of computational modeling in Electroencephalography (EEG) and outline how these models can be used to integrate findings from electrophysiology, network-level models, and behavior. On the one hand, computational models serve to investigate the mechanisms that generate brain activity, for example measured with EEG, such as the transient emergence of oscillations at different frequency bands and/or with different spatial topographies. On the other hand, computational models serve to design experiments and test hypotheses in silico. The final purpose of computational models of EEG is to obtain a comprehensive understanding of the mechanisms that underlie the EEG signal. This is crucial for an accurate interpretation of EEG measurements that may ultimately serve in the development of novel clinical applications.

Algebraic relationship between the structural network's Laplacian and functional network's adjacency matrix is preserved in temporal lobe epilepsy subjects

The relationship between anatomic and resting state functional connectivity of large-scale brain networks is a major focus of current research. In previous work, we introduced a model based on eigen decomposition of the Laplacian which predicts the functional network from the structural network in healthy brains. In this work, we apply the eigen decomposition model to two types of epilepsy; temporal lobe epilepsy associated with mesial temporal sclerosis, and MRI-normal temporal lobe epilepsy. Our findings show that the eigen relationship between function and structure holds for patients with temporal lobe epilepsy as well as normal individuals. These results suggest that the brain under TLE conditions reconfigures and rewires the fine-scale connectivity (a process which the model parameters are putatively sensitive to), in order to achieve the necessary structure-function relationship.

Revisiting correlation-based functional connectivity and its relationship with structural connectivity

Patterns of brain structural connectivity (SC) and functional connectivity (FC) are known to be related. In SC-FC comparisons, FC has classically been evaluated from correlations between functional time series, and more recently from partial correlations or their unnormalized version encoded in the precision matrix. The latter FC metrics yield more meaningful comparisons to SC because they capture ‘direct’ statistical dependencies, that is, discarding the effects of mediators, but their use has been limited because of estimation issues. With the rise of high-quality and large neuroimaging datasets, we revisit the relevance of different FC metrics in the context of SC-FC comparisons. Using data from 100 unrelated Human Connectome Project subjects, we first explore the amount of functional data required to reliably estimate various FC metrics. We find that precision-based FC yields a better match to SC than correlation-based FC when using 5 minutes of functional data or more. Finally, using a linear model linking SC and FC, we show that the SC-FC match can be used to further interrogate various aspects of brain structure and function such as the timescales of functional dynamics in different resting-state networks or the intensity of anatomical self-connections.

Importance of self-connections for brain connectivity and spectral connectomics

Spectral analysis and neural field theory are used to investigate the role of local connections in brain connectivity matrices (CMs) that quantify connectivity between pairs of discretized brain regions. This work investigates how the common procedure of omitting such self-connections (i.e., the diagonal elements of CMs) in published studies of brain connectivity affects the properties of functional CMs (fCMs) and the mutually consistent effective CMs (eCMs) that correspond to them. It is shown that retention of self-connections in the fCM calculated from two-point activity covariances is essential for the fCM to be a true covariance matrix, to enable correct inference of the direct total eCMs from the fCM, and to ensure their compatibility with it; the deCM and teCM represent the strengths of direct connections and all connections between points, respectively. When self-connections are retained, inferred eCMs are found to have net inhibitory self-connections that represent the local inhibition needed to balance excitation via white matter fibers at longer ranges. This inference of spatially unresolved connectivity exemplifies the power of spectral connectivity methods, which also enable transformation of CMs to compact diagonal forms that allow accurate approximation of the fCM and total eCM in terms of just a few modes, rather than the full [Formula: see text] CM entries for connections between N brain regions. It is found that omission of fCM self-connections affects both local and long-range connections in eCMs, so they cannot be omitted even when studying the large-scale. Moreover, retention of local connections enables inference of subgrid short-range inhibitory connectivity. The results are verified and illustrated using the NKI-Rockland dataset from the University of Southern California Multimodal Connectivity Database. Deletion of self-connections is common in the field; this does not affect case-control studies but the present results imply that such fCMs must have self-connections restored before eCMs can be inferred from them.

Path-dependent connectivity, not modularity, consistently predicts controllability of structural brain networks

The human brain displays rich communication dynamics that are thought to be particularly well-reflected in its marked community structure. Yet, the precise relationship between community structure in structural brain networks and the communication dynamics that can emerge therefrom is not well understood. In addition to offering insight into the structure-function relationship of networked systems, such an understanding is a critical step toward the ability to manipulate the brain's large-scale dynamical activity in a targeted manner. We investigate the role of community structure in the controllability of structural brain networks. At the region level, we find that certain network measures of community structure are sometimes statistically correlated with measures of linear controllability. However, we then demonstrate that this relationship depends on the distribution of network edge weights. We highlight the complexity of the relationship between community structure and controllability by performing numerical simulations using canonical graph models with varying mesoscale architectures and edge weight distributions. Finally, we demonstrate that weighted subgraph centrality, a measure rooted in the graph spectrum, and which captures higher order graph architecture, is a stronger and more consistent predictor of controllability. Our study contributes to an understanding of how the brain's diverse mesoscale structure supports transient communication dynamics.

Spectral graph theory of brain oscillations

The relationship between the brain's structural wiring and the functional patterns of neural activity is of fundamental interest in computational neuroscience. We examine a hierarchical, linear graph spectral model of brain activity at mesoscopic and macroscopic scales. The model formulation yields an elegant closed-form solution for the structure-function problem, specified by the graph spectrum of the structural connectome's Laplacian, with simple, universal rules of dynamics specified by a minimal set of global parameters. The resulting parsimonious and analytical solution stands in contrast to complex numerical simulations of high dimensional coupled nonlinear neural field models. This spectral graph model accurately predicts spatial and spectral features of neural oscillatory activity across the brain and was successful in simultaneously reproducing empirically observed spatial and spectral patterns of alpha-band (8-12 Hz) and beta-band (15-30 Hz) activity estimated from source localized magnetoencephalography (MEG). This spectral graph model demonstrates that certain brain oscillations are emergent properties of the graph structure of the structural connectome and provides important insights towards understanding the fundamental relationship between network topology and macroscopic whole-brain dynamics. .

Control of brain network dynamics across diverse scales of space and time

The human brain is composed of distinct regions that are each associated with particular functions and distinct propensities for the control of neural dynamics. However, the relation between these functions and control profiles is poorly understood, as is the variation in this relation across diverse scales of space and time. Here we probe the relation between control and dynamics in brain networks constructed from diffusion tensor imaging data in a large community sample of young adults. Specifically, we probe the control properties of each brain region and investigate their relationship with dynamics across various spatial scales using the Laplacian eigenspectrum. In addition, through analysis of regional modal controllability and partitioning of modes, we determine whether the associated dynamics are fast or slow, as well as whether they are alternating or monotone. We find that brain regions that facilitate the control of energetically easy transitions are associated with activity on short length scales and slow timescales. Conversely, brain regions that facilitate control of difficult transitions are associated with activity on long length scales and fast timescales. Built on linear dynamical models, our results offer parsimonious explanations for the activity propagation and network control profiles supported by regions of differing neuroanatomical structure.

A practical guide to methodological considerations in the controllability of structural brain networks

OBJECTIVE

Predicting how the brain can be driven to specific states by means of internal or external control requires a fundamental understanding of the relationship between neural connectivity and activity. Network control theory is a powerful tool from the physical and engineering sciences that can provide insights regarding that relationship; it formalizes the study of how the dynamics of a complex system can arise from its underlying structure of interconnected units.

APPROACH

Given the recent use of network control theory in neuroscience, it is now timely to offer a practical guide to methodological considerations in the controllability of structural brain networks. Here we provide a systematic overview of the framework, examine the impact of modeling choices on frequently studied control metrics, and suggest potentially useful theoretical extensions. We ground our discussions, numerical demonstrations, and theoretical advances in a dataset of high-resolution diffusion imaging with 730 diffusion directions acquired over approximately 1 h of scanning from ten healthy young adults.

MAIN RESULTS

Following a didactic introduction of the theory, we probe how a selection of modeling choices affects four common statistics: average controllability, modal controllability, minimum control energy, and optimal control energy. Next, we extend the current state-of-the-art in two ways: first, by developing an alternative measure of structural connectivity that accounts for radial propagation of activity through abutting tissue, and second, by defining a complementary metric quantifying the complexity of the energy landscape of a system. We close with specific modeling recommendations and a discussion of methodological constraints.

SIGNIFICANCE

Our hope is that this accessible account will inspire the neuroimaging community to more fully exploit the potential of network control theory in tackling pressing questions in cognitive, developmental, and clinical neuroscience.

Deriving pairwise transfer entropy from network structure and motifs

Transfer entropy (TE) is an established method for quantifying directed statistical dependencies in neuroimaging and complex systems datasets. The pairwise (or bivariate) TE from a source to a target node in a network does not depend solely on the local source-target link weight, but on the wider network structure that the link is embedded in. This relationship is studied using a discrete-time linearly coupled Gaussian model, which allows us to derive the TE for each link from the network topology. It is shown analytically that the dependence on the directed link weight is only a first approximation, valid for weak coupling. More generally, the TE increases with the in-degree of the source and decreases with the in-degree of the target, indicating an asymmetry of information transfer between hubs and low-degree nodes. In addition, the TE is directly proportional to weighted motif counts involving common parents or multiple walks from the source to the target, which are more abundant in networks with a high clustering coefficient than in random networks. Our findings also apply to Granger causality, which is equivalent to TE for Gaussian variables. Moreover, similar empirical results on random Boolean networks suggest that the dependence of the TE on the in-degree extends to nonlinear dynamics.

Model-based whole-brain effective connectivity to study distributed cognition in health and disease

Neuroimaging techniques are now widely used to study human cognition. The functional associations between brain areas have become a standard proxy to describe how cognitive processes are distributed across the brain network. Among the many analysis tools available, dynamic models of brain activity have been developed to overcome the limitations of original connectivity measures such as functional connectivity. This goes in line with the many efforts devoted to the assessment of directional interactions between brain areas from the observed neuroimaging activity. This opinion article provides an overview of our model-based whole-brain effective connectivity to analyze fMRI data, while discussing the pros and cons of our approach with respect to other established approaches. Our framework relies on the multivariate Ornstein-Uhlenbeck (MOU) process and is thus referred to as MOU-EC. Once tuned, the model provides a directed connectivity estimate that reflects the dynamical state of BOLD activity, which can be used to explore cognition. We illustrate this approach using two applications on task-evoked fMRI data. First, as a connectivity measure, MOU-EC can be used to extract biomarkers for task-specific brain coordination, understood as the patterns of areas exchanging information. The multivariate nature of connectivity measures raises several challenges for whole-brain analysis, for which machine-learning tools present some advantages over statistical testing. Second, we show how to interpret changes in MOU-EC connections in a collective and model-based manner, bridging with network analysis. Our framework provides a comprehensive set of tools that open exciting perspectives to study distributed cognition, as well as neuropathologies.

Structural mediation of human brain activity revealed by white-matter interpolation of fMRI

Understanding how the anatomy of the human brain constrains and influences the formation of large-scale functional networks remains a fundamental question in neuroscience. Here, given measured brain activity in gray matter, we interpolate these functional signals into the white matter on a structurally-informed high-resolution voxel-level brain grid. The interpolated volumes reflect the underlying anatomical information, revealing white matter structures that mediate the interaction between temporally coherent gray matter regions. Functional connectivity analyses of the interpolated volumes reveal an enriched picture of the default mode network (DMN) and its subcomponents, including the different white matter bundles that are implicated in their formation, thus extending currently known spatial patterns that are limited within the gray matter only. These subcomponents have distinct structure-function patterns, each of which are differentially observed during tasks, demonstrating plausible structural mechanisms for functional switching between task-positive and -negative components. This work opens new avenues for the integration of brain structure and function, and demonstrates the collective mediation of white matter pathways across short and long-distance functional connections.

Resting-state EEG topographies: Reliable and sensitive signatures of unilateral spatial neglect

Theoretical advances in the neurosciences are leading to the development of an increasing number of proposed interventions for the enhancement of functional recovery after brain damage. Integration of these novel approaches in clinical practice depends on the availability of reliable, simple, and sensitive biomarkers of impairment level and extent of recovery, to enable an informed clinical-decision process. However, the neuropsychological tests currently in use do not tap into the complex neural re-organization process that occurs after brain insult and its modulation by treatment. Here we show that topographical analysis of resting-state electroencephalography (rsEEG) patterns using singular value decomposition (SVD) could be used to capture these processes. In two groups of subacute stroke patients, we show reliable detection of deviant neurophysiological patterns over repeated measurement sessions on separate days. These patterns generalized across patients groups. Additionally, they maintained a significant association with ipsilesional attention bias, discriminating patients with spatial neglect of different severity levels. The sensitivity and reliability of these rsEEG topographical analyses support their use as a tool for monitoring natural and treatment-induced recovery in the rehabilitation process.

Delayed correlations improve the reconstruction of the brain connectome

The brain works as a large-scale complex network, known as the connectome. The strength of the connections between two brain regions in the connectome is commonly estimated by calculating the correlations between their patterns of activation. This approach relies on the assumption that the activation of connected regions occurs together and at the same time. However, there are delays between the activation of connected regions due to excitatory and inhibitory connections. Here, we propose a method to harvest this additional information and reconstruct the structural brain connectome using delayed correlations. This delayed-correlation method correctly identifies 70% to 80% of connections of simulated brain networks, compared to only 5% to 25% of connections detected by the standard methods; this result is robust against changes in the network parameters (small-worldness, excitatory vs. inhibitory connection ratio, weight distribution) and network activation dynamics. The delayed-correlation method predicts more accurately both the global network properties (characteristic path length, global efficiency, clustering coefficient, transitivity) and the nodal network properties (nodal degree, nodal clustering, nodal global efficiency), particularly at lower network densities. We obtain similar results in networks derived from animal and human data. These results suggest that the use of delayed correlations improves the reconstruction of the structural brain connectome and open new possibilities for the analysis of the brain connectome, as well as for other types of networks.

Heritability and Cognitive Relevance of Structural Brain Controllability

Cognition and behavior are thought to emerge from the connections and interactions among brain regions. The precise nature of these relationships remains elusive. Here we use tools provided by network control theory to determine how the structural connectivity profile of brain regions may shape individual variation in cognition. In a cohort of healthy young adults (n = 1066), we computed two fundamental brain regional control patterns, average and modal controllability, which index the degree of influence of a region over others. We first established that regional brain controllability measures were both reproducible and heritable. Regions with controllability profiles theoretically conducive to facilitating multiple cognitive operations were over-represented in higher-order resting-state networks. Finally, variation in regional controllability accounted for about 50% of interindividual variability in multiple cognitive domains. We conclude that controllability is a biologically plausible property of the structural connectome and provides a mechanistic explanation for how brain structural architecture may influence cognitive functions.

Longitudinal increases in structural connectome segregation and functional connectome integration are associated with better recovery after mild TBI

Traumatic brain injury damages white matter pathways that connect brain regions, disrupting transmission of electrochemical signals and causing cognitive and emotional dysfunction. Connectome-level mechanisms for how the brain compensates for injury have not been fully characterized. Here, we collected serial MRI-based structural and functional connectome metrics and neuropsychological scores in 26 mild traumatic brain injury subjects (29.4 ± 8.0 years, 20 males) at 1 and 6 months postinjury. We quantified the relationship between functional and structural connectomes using network diffusion (ND) model propagation time, a measure that can be interpreted as how much of the structural connectome is being utilized for the spread of functional activation, as captured via the functional connectome. Overall cognition showed significant improvement from 1 to 6 months (t25 = -2.15, p = .04). None of the structural or functional global connectome metrics was significantly different between 1 and 6 months, or when compared to 34 age- and gender-matched controls (28.6 ± 8.8 years, 25 males). We predicted longitudinal changes in overall cognition from changes in global connectome measures using a partial least squares regression model (cross-validated R2 = .27). We observe that increased ND model propagation time, increased structural connectome segregation, and increased functional connectome integration were related to better cognitive recovery. We interpret these findings as suggesting two connectome-based postinjury recovery mechanisms: one of neuroplasticity that increases functional connectome integration and one of remote white matter degeneration that increases structural connectome segregation. We hypothesize that our inherently multimodal measure of ND model propagation time captures the interplay between these two mechanisms.

The role of network structure and time delay in a metapopulation Wilson--Cowan model

We study the dynamics of a network Wilson--Cowan model (a system of connected Wilson--Cowan oscillators) for interacting excitatory and inhibitory neuron populations with time delays. Each node in this model corresponds to a population of neurons, including excitatory and inhibitory subpopulations, and hence it can be viewed as a metapopulation model. It is known that information transfer within each cortical area is not instantaneous, and therefore we consider a system of delay differential equations with two different kinds of discrete delays. We account for the time delay in information propagation between individual excitatory and inhibitory subpopulations at each node via intra-node time delays, and we account for time delay in information propagation between neuron populations at different nodes with inter-node time delays. The biologically relevant resting state solutions are oscillatory (stable limit cycles). After determining the influence of the coupling parameters between nodes, the intra-node delays, and the inter-node delays on the dynamics of the two coupled Wilson--Cowan oscillators, we then explore a variety of larger networks of 16 and 100 nodes, in order to determine how the network topology will influence time delayed Wilson--Cowan dynamics. We find that network structure can regularize or deregularize the dynamics, with networks of higher mean degree permitting stable limit cycles and networks with smaller mean degree yielding less regular dynamics (which may range from chaotic solutions, to solutions for which limit cycles collapse into steady states, which are biologically undesirable compared with the preferred stable limit cycles). Furthermore, heterogeneity in the degree distribution of the network (resulting from networks with nodes of varying degree) can result in asynchronous dynamics, even if at each node the local dynamics are that of a limit cycle, in contrast to the synchronization of dynamics between nodes seen when the degree of all nodes is equal. This suggests that homogeneous and well-connected networks permit robust limit cycles under time-delayed Wilson--Cowan dynamics, whereas heterogeneous or poorly connected networks may fail to provide such desirable dynamics, a phenomena akin to structural loss of neuron connections in neurodegenerative diseases.

Interhemispheric Synchrony of Spontaneous Cortical States at the Cortical Column Level

In cat early visual cortex, neural activity patterns resembling evoked orientation maps emerge spontaneously under anesthesia. To test if such patterns are synchronized between hemispheres, we performed bilateral imaging in anesthetized cats using a new improved voltage-sensitive dye. We observed map-like activity patterns spanning early visual cortex in both hemispheres simultaneously. Patterns virtually identical to maps associated with the cardinal and oblique orientations emerged as leading principal components of the spontaneous fluctuations, and the strength of transient orientation states was correlated with their duration, providing evidence that these maps are transiently attracting states. A neural mass model we developed reproduced the dynamics of both smooth and abrupt orientation state transitions observed experimentally. The model suggests that map-like activity arises from slow modulations in spontaneous firing in conjunction with interplay between excitation and inhibition. Our results highlight the efficiency and functional precision of interhemispheric connectivity.

Fluctuation Scaling of Neuronal Firing and Bursting in Spontaneously Active Brain Circuits

We employed high-density microelectrode arrays to investigate spontaneous firing patterns of neurons in brain circuits of the primary somatosensory cortex (S1) in mice. We recorded from over 150 neurons for 10min in each of eight different experiments, identified their location in S1, sorted their action potentials (spikes), and computed their power spectra and inter-spike interval (ISI) statistics. Of all persistently active neurons, 92% fired with a single dominant frequency - regularly firing neurons (RNs) - from 1 to 8Hz while 8% fired in burst with two dominant frequencies - bursting neurons (BNs) - corresponding to the inter-burst (2-6Hz) and intra-burst intervals (20-160Hz). RNs were predominantly located in layers 2/3 and 5/6 while BNs localized to layers 4 and 5. Across neurons, the standard deviation of ISI was a power law of its mean, a property known as fluctuation scaling, with a power law exponent of 1 for RNs and 1.25 for BNs. The power law implies that firing and bursting patterns are scale invariant: the firing pattern of a given RN or BN resembles that of another RN or BN, respectively, after a time contraction or dilation. An explanation for this scale invariance is discussed in the context of previous computational studies as well as its potential role in information processing.

Network analysis of whole-brain fMRI dynamics: A new framework based on dynamic communicability

Neuroimaging techniques such as MRI have been widely used to explore the associations between brain areas. Structural connectivity (SC) captures the anatomical pathways across the brain and functional connectivity (FC) measures the correlation between the activity of brain regions. These connectivity measures have been much studied using network theory in order to uncover the distributed organization of brain structures, in particular FC for task-specific brain communication. However, the application of network theory to study FC matrices is often "static" despite the dynamic nature of time series obtained from fMRI. The present study aims to overcome this limitation by introducing a network-oriented analysis applied to whole-brain effective connectivity (EC) useful to interpret the brain dynamics. Technically, we tune a multivariate Ornstein-Uhlenbeck (MOU) process to reproduce the statistics of the whole-brain resting-state fMRI signals, which provides estimates for MOU-EC as well as input properties (similar to local excitabilities). The network analysis is then based on the Green function (or network impulse response) that describes the interactions between nodes across time for the estimated dynamics. This model-based approach provides time-dependent graph-like descriptor, named communicability, that characterize the roles that either nodes or connections play in the propagation of activity within the network. They can be used at both global and local levels, and also enables the comparison of estimates from real data with surrogates (e.g. random network or ring lattice). In contrast to classical graph approaches to study SC or FC, our framework stresses the importance of taking the temporal aspect of fMRI signals into account. Our results show a merging of functional communities over time, moving from segregated to global integration of the network activity. Our formalism sets a solid ground for the analysis and interpretation of fMRI data, including task-evoked activity.

Brain controllability: Not a slam dunk yet

In our recent article [1] published in this journal we provide quantitative evidence to show that there are warnings and caveats in the way Gu and collaborators [2] define controllability of brain networks and measure the contribution of each of its nodes. The comment by Pasqualetti et al. [3] confirms the need to go beyond the methodology and approach presented in Gu et al.'s original work. In fact, they recognize that "the source of confusion is due to the fact that assessing controllability via numerical analysis typically leads to ill-conditioned problems, and thus often generates results that are difficult to interpret". This is indeed the first warning we discussed in [1]: our work was not meant to prove that brain networks are not controllable from one node, rather we wished to highlight that the one node controllability framework and all consequent results were not properly justified based on the methodology presented in Gu et al. [2]. We used in our work the same method of Gu et al. not because we believe it is the best methodology, but because we extensively investigated it with the aim of replicating, testing, and extending their results. The warning and caveats we have proposed are the results of this investigation. Indeed, on the basis of our controllability analyses of multiple human brain networks datasets, we concluded: "The λ_min(W_K) are statistically compatible with zero and thus the associated controllability Gramian cannot be inverted¹. These results show that it is not possible to infer one node controllability of the brain numerically". Hence both groups agree that one node controllability cannot be inferred numerically.

Determination of effective brain connectivity from activity correlations

Effective connectivity embodied in transfer functions is derived from symmetric-network activity correlations under task-free conditions via a recent causal spectral factorization method. This generalizes previous covariance-based analyses to include frequency dependencies and time delays. Results are verified against analytic solutions of equations that have reproduced many aspects of experimental brain dynamics and against cases of more complex connectivity. Robustness to noise is also demonstrated.

Neural field theory of effects of brain modifications and lesions on functional connectivity: Acute effects, short-term homeostasis, and long-term plasticity

Neural field theory is used to predict the functional connectivity effects of lesions or other modifications to effective connectivity. Widespread initial changes are predicted after localized or diffuse changes to white or gray matter, consistent with observations, and enabling lesion severity indexes to be defined. It is shown how short-term homeostasis and longer-term plasticity can reduce perturbations while maintaining brain criticality under conditions where some connections remain fixed because of damage in the lesion core. The extent to which such effects can compensate for initial connectivity changes is then explored, showing that the strongest corrective changes are concentrated toward the edges of the perturbation if it is localized and its core is fixed. The results are applicable to inferring underlying connectivity changes and to interpreting and monitoring functional connectivity modifications after lesions, injury, surgery, drugs, or brain stimulation.

Physical brain connectomics

Brain connectivity and structure-function relationships are analyzed from a physical perspective in place of common graph-theoretic and statistical approaches that overwhelmingly ignore the brain's physical structure and geometry. Field theory is used to define connectivity tensors in terms of bare and dressed propagators, and discretized representations are implemented that respect the physical nature and dimensionality of the quantities involved, retain the correct continuum limit, and enable diagrammatic analysis. Eigenfunction analysis is used to simultaneously characterize and probe patterns of brain connectivity and activity, in place of statistical or phenomenological patterns. Physically based measures that characterize the connectivity are then developed in coordinate and spectral domains; some of which generalize or rectify graph-theoretic measures to implement correct dimensionality and continuum limits, and some replace graph-theoretic quantities. Traditional graph-based measures are shown to be highly prone to artifacts introduced by discretization and threshold, often because essential physical constraints have not been imposed, dimensionality has not been included, and/or distinctions between scalar, vector, and tensor quantities have not been considered. The results can replace them in ways that converge correctly and measure properties of brain structure, rather than of its discretization, and thus potentially enable physical interpretation of the many phenomenological results in the literature. Geometric effects are shown to dominate in determining many brain properties and care must be taken not to interpret geometric differences as differences in intrinsic neural connectivity. The results demonstrate the need to use systematic physical methods to analyze the brain and the potential of such methods to obtain new insights from data, make new predictions for experimental test, and go beyond phenomenological classification to dynamics and mechanisms.

Medial Prefrontal Cortex Population Activity Is Plastic Irrespective of Learning

The prefrontal cortex (PFC) is thought to learn the relationships between actions and their outcomes. But little is known about what changes to population activity in PFC are specific to learning these relationships. Here we characterize the plasticity of population activity in the medial PFC (mPFC) of male rats learning rules on a Y-maze. First, we show that the population always changes its patterns of joint activity between the periods of sleep either side of a training session on the maze, regardless of successful rule learning during training. Next, by comparing the structure of population activity in sleep and training, we show that this population plasticity differs between learning and nonlearning sessions. In learning sessions, the changes in population activity in post-training sleep incorporate the changes to the population activity during training on the maze. In nonlearning sessions, the changes in sleep and training are unrelated. Finally, we show evidence that the nonlearning and learning forms of population plasticity are driven by different neuron-level changes, with the nonlearning form entirely accounted for by independent changes to the excitability of individual neurons, and the learning form also including changes to firing rate couplings between neurons. Collectively, our results suggest two different forms of population plasticity in mPFC during the learning of action-outcome relationships: one a persistent change in population activity structure decoupled from overt rule-learning, and the other a directional change driven by feedback during behavior.SIGNIFICANCE STATEMENT The PFC is thought to represent our knowledge about what action is worth doing in which context. But we do not know how the activity of neurons in PFC collectively changes when learning which actions are relevant. Here we show, in a trial-and-error task, that population activity in PFC is persistently changing, regardless of learning. Only during episodes of clear learning of relevant actions are the accompanying changes to population activity carried forward into sleep, suggesting a long-lasting form of neural plasticity. Our results suggest that representations of relevant actions in PFC are acquired by reward imposing a direction onto ongoing population plasticity.

Distributed network interactions and their emergence in developing neocortex

The principles governing the functional organization and development of long-range network interactions in the neocortex remain poorly understood. Using in vivo wide-field and 2-photon calcium imaging of spontaneous activity patterns in mature ferret visual cortex, we find widespread modular correlation patterns that accurately predict the local structure of visually-evoked orientation columns several millimeters away. Longitudinal imaging demonstrates that long-range spontaneous correlations are present early in cortical development prior to the elaboration of horizontal connections, and predict mature network structure. Silencing feed-forward drive through retinal or thalamic blockade does not eliminate early long-range correlated activity, suggesting a cortical origin. Circuit models containing only local, but heterogeneous, connections are sufficient to generate long-range correlated activity by confining activity patterns to a low-dimensional subspace via multi-synaptic short-range interactions. These results suggest that local connections in early cortical circuits can generate structured long-range network correlations that guide the formation of visually-evoked distributed functional networks.

A computational network control theory analysis of depression symptoms

Rumination and impaired inhibition are considered core characteristics of depression. However, the neurocognitive mechanisms that contribute to these atypical cognitive processes remain unclear. To address this question, we apply a computational network control theory approach to structural brain imaging data acquired via diffusion tensor imaging in a large sample of participants, to examine how network control theory relates to individual differences in subclinical depression. Recent application of this theory at the neural level is built on a model of brain dynamics, which mathematically models patterns of inter-region activity propagated along the structure of an underlying network. The strength of this approach is its ability to characterize the potential role of each brain region in regulating whole-brain network function based on its anatomical fingerprint and a simplified model of node dynamics. We find that subclinical depression is negatively related to higher integration abilities in the right anterior insula, replicating and extending previous studies implicating atypical switching between the default mode and Executive Control Networks in depression. We also find that subclinical depression is related to the ability to “drive” the brain system into easy to reach neural states in several brain regions, including the bilateral lingual gyrus and lateral occipital gyrus. These findings highlight brain regions less known in their role in depression, and clarify their roles in driving the brain into different neural states related to depression symptoms.

Driving the brain towards creativity and intelligence: A network control theory analysis

High-level cognitive constructs, such as creativity and intelligence, entail complex and multiple processes, including cognitive control processes. Recent neurocognitive research on these constructs highlight the importance of dynamic interaction across neural network systems and the role of cognitive control processes in guiding such a dynamic interaction. How can we quantitatively examine the extent and ways in which cognitive control contributes to creativity and intelligence? To address this question, we apply a computational network control theory (NCT) approach to structural brain imaging data acquired via diffusion tensor imaging in a large sample of participants, to examine how NCT relates to individual differences in distinct measures of creative ability and intelligence. Recent application of this theory at the neural level is built on a model of brain dynamics, which mathematically models patterns of inter-region activity propagated along the structure of an underlying network. The strength of this approach is its ability to characterize the potential role of each brain region in regulating whole-brain network function based on its anatomical fingerprint and a simplified model of node dynamics. We find that intelligence is related to the ability to "drive" the brain system into easy to reach neural states by the right inferior parietal lobe and lower integration abilities in the left retrosplenial cortex. We also find that creativity is related to the ability to "drive" the brain system into difficult to reach states by the right dorsolateral prefrontal cortex (inferior frontal junction) and higher integration abilities in sensorimotor areas. Furthermore, we found that different facets of creativity-fluency, flexibility, and originality-relate to generally similar but not identical network controllability processes. We relate our findings to general theories on intelligence and creativity.

Warnings and caveats in brain controllability

A recent article by Gu et al. (Nat. Commun. 6, 2015) proposed to characterize brain networks, quantified using anatomical diffusion imaging, in terms of their "controllability", drawing on concepts and methods of control theory. They reported that brain activity is controllable from a single node, and that the topology of brain networks provides an explanation for the types of control roles that different regions play in the brain. In this work, we first briefly review the framework of control theory applied to complex networks. We then show contrasting results on brain controllability through the analysis of five different datasets and numerical simulations. We find that brain networks are not controllable (in a statistical significant way) by one single region. Additionally, we show that random null models, with no biological resemblance to brain network architecture, produce the same type of relationship observed by Gu et al. between the average/modal controllability and weighted degree. Finally, we find that resting state networks defined with fMRI cannot be attributed specific control roles. In summary, our study highlights some warning and caveats in the brain controllability framework.

Specificity and robustness of long-distance connections in weighted, interareal connectomes

Brain areas' functional repertoires are shaped by their incoming and outgoing structural connections. In empirically measured networks, most connections are short, reflecting spatial and energetic constraints. Nonetheless, a small number of connections span long distances, consistent with the notion that the functionality of these connections must outweigh their cost. While the precise function of long-distance connections is unknown, the leading hypothesis is that they act to reduce the topological distance between brain areas and increase the efficiency of interareal communication. However, this hypothesis implies a nonspecificity of long-distance connections that we contend is unlikely. Instead, we propose that long-distance connections serve to diversify brain areas' inputs and outputs, thereby promoting complex dynamics. Through analysis of five weighted interareal network datasets, we show that long-distance connections play only minor roles in reducing average interareal topological distance. In contrast, areas' long-distance and short-range neighbors exhibit marked differences in their connectivity profiles, suggesting that long-distance connections enhance dissimilarity between areal inputs and outputs. Next, we show that-in isolation-areas' long-distance connectivity profiles exhibit nonrandom levels of similarity, suggesting that the communication pathways formed by long connections exhibit redundancies that may serve to promote robustness. Finally, we use a linearization of Wilson-Cowan dynamics to simulate the covariance structure of neural activity and show that in the absence of long-distance connections a common measure of functional diversity decreases. Collectively, our findings suggest that long-distance connections are necessary for supporting diverse and complex brain dynamics.

Functional brain connectivity is predictable from anatomic network's Laplacian eigen-structure

How structural connectivity (SC) gives rise to functional connectivity (FC) is not fully understood. Here we mathematically derive a simple relationship between SC measured from diffusion tensor imaging, and FC from resting state fMRI. We establish that SC and FC are related via (structural) Laplacian spectra, whereby FC and SC share eigenvectors and their eigenvalues are exponentially related. This gives, for the first time, a simple and analytical relationship between the graph spectra of structural and functional networks. Laplacian eigenvectors are shown to be good predictors of functional eigenvectors and networks based on independent component analysis of functional time series. A small number of Laplacian eigenmodes are shown to be sufficient to reconstruct FC matrices, serving as basis functions. This approach is fast, and requires no time-consuming simulations. It was tested on two empirical SC/FC datasets, and was found to significantly outperform generative model simulations of coupled neural masses.

Framework based on communicability and flow to analyze complex network dynamics

Graph theory constitutes a widely used and established field providing powerful tools for the characterization of complex networks. The intricate topology of networks can also be investigated by means of the collective dynamics observed in the interactions of self-sustained oscillations (synchronization patterns) or propagationlike processes such as random walks. However, networks are often inferred from real-data-forming dynamic systems, which are different from those employed to reveal their topological characteristics. This stresses the necessity for a theoretical framework dedicated to the mutual relationship between the structure and dynamics in complex networks, as the two sides of the same coin. Here we propose a rigorous framework based on the network response over time (i.e., Green function) to study interactions between nodes across time. For this purpose we define the flow that describes the interplay between the network connectivity and external inputs. This multivariate measure relates to the concepts of graph communicability and the map equation. We illustrate our theory using the multivariate Ornstein-Uhlenbeck process, which describes stable and non-conservative dynamics, but the formalism can be adapted to other local dynamics for which the Green function is known. We provide applications to classical network examples, such as small-world ring and hierarchical networks. Our theory defines a comprehensive framework that is canonically related to directed and weighted networks, thus paving a way to revise the standards for network analysis, from the pairwise interactions between nodes to the global properties of networks including community detection.

Neural field theory of perceptual echo and implications for estimating brain connectivity

Neural field theory is used to predict and analyze the phenomenon of perceptual echo in which random input stimuli at one location are correlated with electroencephalographic responses at other locations. It is shown that this echo correlation (EC) yields an estimate of the transfer function from the stimulated point to other locations. Modal analysis then explains the observed spatiotemporal structure of visually driven EC and the dominance of the alpha frequency; two eigenmodes of similar amplitude dominate the response, leading to temporal beating and a line of low correlation that runs from the crown of the head toward the ears. These effects result from mode splitting and symmetry breaking caused by interhemispheric coupling and cortical folding. It is shown how eigenmodes obtained from functional magnetic resonance imaging experiments can be combined with temporal dynamics from EC or other evoked responses to estimate the spatiotemporal transfer function between any two points and hence their effective connectivity.

Multiple Kernel Learning Model for Relating Structural and Functional Connectivity in the Brain

A challenging problem in cognitive neuroscience is to relate the structural connectivity (SC) to the functional connectivity (FC) to better understand how large-scale network dynamics underlying human cognition emerges from the relatively fixed SC architecture. Recent modeling attempts point to the possibility of a single diffusion kernel giving a good estimate of the FC. We highlight the shortcomings of the single-diffusion-kernel model (SDK) and propose a multi-scale diffusion scheme. Our multi-scale model is formulated as a reaction-diffusion system giving rise to spatio-temporal patterns on a fixed topology. We hypothesize the presence of inter-regional co-activations (latent parameters) that combine diffusion kernels at multiple scales to characterize how FC could arise from SC. We formulated a multiple kernel learning (MKL) scheme to estimate the latent parameters from training data. Our model is analytically tractable and complex enough to capture the details of the underlying biological phenomena. The parameters learned by the MKL model lead to highly accurate predictions of subject-specific FCs from test datasets at a rate of 71%, surpassing the performance of the existing linear and non-linear models. We provide an example of how these latent parameters could be used to characterize age-specific reorganization in the brain structure and function.