Determination of effective brain connectivity from functional connectivity with application to resting state connectivities

Near-critical corticothalamic eigenmodes: Effects of nonuniform connectivity on modes, activity, and communication channels

The effects of nonuniformities in axonal connectivity on natural modes of brain activity are explored to determine their contributions to modal eigenvalues, structure, and communication and to clarify the limits of validity of widely used uniform-connectivity approximations. Preferred channels of communication are demonstrated that are supported by natural modes of mean connectivity and resulting activity. The effects of axonal tracts on these modes are calculated using perturbation methods, and it is found that modes and their spectra are only moderately perturbed by even the largest white matter tracts. However, perturbations of activity are greatly magnified when modes are near-critical and realistic connectivity and gain perturbations can then enable rapid responses to stimuli on the observed timescales of evoked responses. It is thus argued that dynamic mode-mode communication channels complement ones based on white matter tracts and that both rely on near-criticality to have their observed effects.

Unified theory of alpha, mu, and tau rhythms via eigenmodes of brain activity

A compact description of the frequency structure and topography of human alpha-band rhythms is obtained by use of the first four brain activity eigenmodes previously derived from corticothalamic neural field theory. Just two eigenmodes that overlap in frequency are found to reproduce the observed topography of the classical alpha rhythm for subjects with a single, occipitally concentrated alpha peak in their electroencephalograms. Alpha frequency splitting and relative amplitudes of double alpha peaks are explored analytically and numerically within this four-mode framework using eigenfunction expansion and perturbation methods. These effects are found to result primarily from the different eigenvalues and corticothalamic gains corresponding to the eigenmodes. Three modes with two non-overlapping frequencies suffice to reproduce the observed topography for subjects with a double alpha peak, where the appearance of a distinct second alpha peak requires an increase of the corticothalamic gain of higher eigenmodes relative to the first. Conversely, alpha blocking is inferred to be linked to a relatively small attention-dependent reduction of the gain of the relevant eigenmodes, whose effect is enhanced by the near-critical state of the brain and whose sign is consistent with inferences from neural field theory. The topographies and blocking of the mu and tau rhythms within the alpha-band are explained analogously via eigenmodes. Moreover, the observation of three rhythms in the alpha band is due to there being exactly three members of the first family of spatially nonuniform modes. These results thus provide a simple, unified description of alpha band rhythms and enable experimental observations of spectral structure and topography to be linked directly to theory and underlying physiology.

Decoding Autism: Uncovering patterns in brain connectivity through sparsity analysis with rs-fMRI data

BACKGROUND

In the realm of neuro-disorders, precise diagnosis and treatment rely heavily on objective imaging-based biomarker identification. This study employs a sparsity approach on resting-state fMRI to discern relevant brain region connectivity for predicting Autism.

NEW METHOD

The proposed methodology involves four key steps: (1) Utilizing three probabilistic brain atlases to extract functionally homogeneous brain regions from fMRI data. (2) Employing a hybrid approach of Graphical Lasso and Akaike Information Criteria to optimize sparse inverse covariance matrices for representing the brain functional connectivity. (3) Employing statistical techniques to scrutinize functional brain structures in Autism and Control subjects. (4) Implementing both autoencoder-based feature extraction and entire feature-based approach coupled with AI-based learning classifiers to predict Autism.

RESULTS

The ensemble classifier with the extracted feature set achieves a classification accuracy of 84.7% ± 0.3% using the MSDL atlas. Meanwhile, the 1D-CNN model, employing all features, exhibits superior classification accuracy of 88.6% ± 1.7% with the Smith 2009 (rsn70) atlas.

COMPARISON WITH EXISTING METHOD (S)

The proposed methodology outperforms the conventional correlation-based functional connectivity approach with a notably high prediction accuracy of more than 88%, whereas considering all direct and noisy indirect region-based functional connectivity, the traditional methods bound the prediction accuracy within 70% to 79%.

CONCLUSIONS

This study underscores the potential of sparsity-based FC analysis using rs-fMRI data as a prognostic biomarker for detecting Autism.

Bridging functional and anatomical neural connectivity through cluster synchronization

The dynamics of the brain results from the complex interplay of several neural populations and is affected by both the individual dynamics of these areas and their connection structure. Hence, a fundamental challenge is to derive models of the brain that reproduce both structural and functional features measured experimentally. Our work combines neuroimaging data, such as dMRI, which provides information on the structure of the anatomical connectomes, and fMRI, which detects patterns of approximate synchronous activity between brain areas. We employ cluster synchronization as a tool to integrate the imaging data of a subject into a coherent model, which reconciles structural and dynamic information. By using data-driven and model-based approaches, we refine the structural connectivity matrix in agreement with experimentally observed clusters of brain areas that display coherent activity. The proposed approach leverages the assumption of homogeneous brain areas; we show the robustness of this approach when heterogeneity between the brain areas is introduced in the form of noise, parameter mismatches, and connection delays. As a proof of concept, we apply this approach to MRI data of a healthy adult at resting state.

Accurately Modeling the Resting Brain Functional Correlations Using Wave Equation With Spatiotemporal Varying Hypergraph Laplacian

How spontaneous brain neural activities emerge from the underlying anatomical architecture, characterized by structural connectivity (SC), has puzzled researchers for a long time. Over the past decades, much effort has been directed toward the graph modeling of SC, in which the brain SC is generally considered as relatively invariant. However, the graph representation of SC is unable to directly describe the connections between anatomically unconnected brain regions and fail to model the negative functional correlations. Here, we extend the static graph model to a spatiotemporal varying hypergraph Laplacian diffusion (STV-HGLD) model to describe the propagation of the spontaneous neural activity in human brain by incorporating the Laplacian of the hypergraph representation of the structural connectome ( h SC) into the regular wave equation. Theoretical solution shows that the dynamic functional couplings between brain regions fluctuate in the form of an exponential wave regulated by the spatiotemporal varying Laplacian of h SC. Empirical study suggests that the cortical wave might give rise to resonance with SC during the self-organizing interplay between excitation and inhibition among brain regions, which orchestrates the cortical waves propagating with harmonics emanating from the h SC while being bound by the natural frequencies of SC. Besides, the average statistical dependencies between brain regions, normally defined as the functional connectivity (FC), arises just at the moment before the cortical wave reaches the steady state after the wave spreads across all the brain regions. Comprehensive tests on four extensively studied empirical brain connectome datasets with different resolutions confirm our theory and findings.

The music of the hemispheres: Cortical eigenmodes as a physical basis for large-scale brain activity and connectivity patterns

Neuroscience has had access to high-resolution recordings of large-scale cortical activity and structure for decades, but still lacks a generally adopted basis to analyze and interrelate results from different individuals and experiments. Here it is argued that the natural oscillatory modes of the cortex-cortical eigenmodes-provide a physically preferred framework for systematic comparisons across experimental conditions and imaging modalities. In this framework, eigenmodes are analogous to notes of a musical instrument, while commonly used statistical patterns parallel frequently played chords. This intuitive perspective avoids problems that often arise in neuroimaging analyses, and connects to underlying mechanisms of brain activity. We envisage this approach will lead to novel insights into whole-brain function, both in existing and prospective datasets, and facilitate a unification of empirical findings across presently disparate analysis paradigms and measurement modalities.

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.

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.

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.

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.

Brain dynamics and structure-function relationships via spectral factorization and the transfer function

It is shown how the brain's linear transfer function provides a means to systematically analyze brain connectivity and dynamics, and to infer connectivity, eigenmodes, and activity measures such as spectra, evoked responses, coherence, and causality, all of which are widely used in brain monitoring. In particular, the Wilson spectral factorization algorithm is outlined and used to efficiently obtain linear transfer functions from experimental two-point correlation functions. The algorithm is tested on a series of brain-like structures of increasing complexity which include time delays, asymmetry, two-dimensionality, and complex network connectivity. These tests are used to verify the algorithm is suitable for application to brain dynamics, specify sampling requirements for experimental time series, and to verify that its runtime is short enough to obtain accurate results for systems of similar size to current experiments. The results can equally well be applied to inference of the transfer function in complex linear systems other than brains.

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.

Mapping functional brain networks from the structural connectome: Relating the series expansion and eigenmode approaches

Functional brain networks are shaped and constrained by the underlying structural network. However, functional networks are not merely a one-to-one reflection of the structural network. Several theories have been put forward to understand the relationship between structural and functional networks. However, it remains unclear how these theories can be unified. Two existing recent theories state that 1) functional networks can be explained by all possible walks in the structural network, which we will refer to as the series expansion approach, and 2) functional networks can be explained by a weighted combination of the eigenmodes of the structural network, the so-called eigenmode approach. To elucidate the unique or common explanatory power of these approaches to estimate functional networks from the structural network, we analysed the relationship between these two existing views. Using linear algebra, we first show that the eigenmode approach can be written in terms of the series expansion approach, i.e., walks on the structural network associated with different hop counts correspond to different weightings of the eigenvectors of this network. Second, we provide explicit expressions for the coefficients for both the eigenmode and series expansion approach. These theoretical results were verified by empirical data from Diffusion Tensor Imaging (DTI) and functional Magnetic Resonance Imaging (fMRI), demonstrating a strong correlation between the mappings based on both approaches. Third, we analytically and empirically demonstrate that the fit of the eigenmode approach to measured functional data is always at least as good as the fit of the series expansion approach, and that errors in the structural data lead to large errors of the estimated coefficients for the series expansion approach. Therefore, we argue that the eigenmode approach should be preferred over the series expansion approach. Results hold for eigenmodes of the weighted adjacency matrices as well as eigenmodes of the graph Laplacian. ​Taken together, these results provide an important step towards unification of existing theories regarding the structure-function relationships in brain networks.

Effective brain connectivity estimation between active brain regions in autism using the dual Kalman-based method

Brain connectivity estimation is a useful method to study brain functions and diagnose neuroscience disorders. Effective connectivity is a subdivision of brain connectivity which discusses the causal relationship between different parts of the brain. In this study, a dual Kalman-based method is used for effective connectivity estimation. Because of connectivity changes in autism, the method is applied to autistic signals for effective connectivity estimation. For method validation, the dual Kalman based method is compared with other connectivity estimation methods by estimation error and the dual Kalman-based method gives acceptable results with less estimation errors. Then, connectivities between active brain regions of autistic and normal children in the resting state are estimated and compared. In this simulation, the brain is divided into eight regions and the connectivity between regions and within them is calculated. It can be concluded from the results that in the resting state condition the effective connectivity of active regions is decreased between regions and is increased within each region in autistic children. In another result, by averaging the connectivity between the extracted active sources of each region, the connectivity between the left and right of the central part is more than that in other regions and the connectivity in the occipital part is less than that in others.

Dependence of connectivity on geometric distance in brain networks

In any network, the dependence of connectivity on physical distance between nodes is a direct consequence of trade-off mechanisms between costs of establishing and sustaining links, processing rates, propagation speed of signals between nodes. Despite its universality, there are still few studies addressing this issue. Here we apply a recently-developed method to infer links between nodes, and possibly subnetwork structures, to determine connectivity strength as a function of physical distance between nodes. The model system we investigate is brain activity reconstructed on the cortex out of magnetoencephalography recordings sampled on a set of healthy subjects in resting state. We found that the dependence of the time scale of observability of a link on its geometric length follows a power-law characterized by an exponent whose extent is inversely proportional to connectivity. Our method provides a new tool to highlight and investigate networks in neuroscience.

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.

Changes in resting-state functional brain activity are associated with waning cognitive functions in HIV-infected children

Delayed brain development in perinatally HIV-infected children may affect the functional brain activity and subsequently cognitive function. The current study evaluated the functional brain activity in HIV-infected children by quantifying the amplitude of low frequency fluctuations (ALFF) and functional connectivity (FC). Additionally, correlation of ALFF and FC with cognitive measures was performed. Twenty-six HIV-infected children and 20 control children underwent neuropsychological (NP) assessment and resting-state functional magnetic resonance imaging (rs-fMRI). ALFF and FC maps were generated and group differences were analyzed using two-sample t-test. Furthermore, ALFF and FC showing significant group differences were correlated with NP scores using Pearson's correlation. Significantly lower ALFF in the left middle temporal gyrus, precentral and post central gyrus was observed in HIV-infected children compared to controls. FC was significantly reduced in the right inferior parietal, vermis, middle temporal and left postcentral regions, and significantly increased in the right precuneus, superior parietal and left middle frontal regions in HIV-infected children as compared to control. HIV-infected children showed significantly lower NP scores in various domains including closure, exclusion, memory, verbal meaning, quantity and hidden figure than controls. These waning cognitive functions were significantly associated with changes in ALFF and FC in HIV-infected children. The findings suggest that abnormal ALFF and FC may responsible for cognitive deficits in HIV-infected children. ALFF and FC in association with cognitive evaluation may provide a clinical biomarker to evaluate functional brain activity and to plan neurocognitive intervention in HIV-infected children undergoing standard treatment.

Spectrum of connectivity fluctuations including the effect of activity-dependent feedback

The spatiotemporal spectrum of feedback-driven fluctuations of brain connectivity is investigated using nonlinear neural field theory of the corticothalamic system. Weakly nonlinear dynamics of neural feedbacks are expanded in terms of first order perturbations of neural activity relative to a fixed point. Susceptibilities are used to quantify the change in connectivity per unit change in presynaptic or postsynaptic activity caused by nonlinear feedbacks such as facilitation, depression, sensitization, potentiation, and the effects of discrete eigenmode structure are included for a spherical brain geometry. Spectral signatures such as resonances are identified that allow the presence of particular presynaptic and postsynaptic feedback effects to be inferred. These include additional resonances at high frequencies and shifts of existing spectral peaks, mostly visible in the lowest spatial modes of the response.

NFTsim: Theory and Simulation of Multiscale Neural Field Dynamics

A user ready, portable, documented software package, NFTsim, is presented to facilitate numerical simulations of a wide range of brain systems using continuum neural field modeling. NFTsim enables users to simulate key aspects of brain activity at multiple scales. At the microscopic scale, it incorporates characteristics of local interactions between cells, neurotransmitter effects, synaptodendritic delays and feedbacks. At the mesoscopic scale, it incorporates information about medium to large scale axonal ranges of fibers, which are essential to model dissipative wave transmission and to produce synchronous oscillations and associated cross-correlation patterns as observed in local field potential recordings of active tissue. At the scale of the whole brain, NFTsim allows for the inclusion of long range pathways, such as thalamocortical projections, when generating macroscopic activity fields. The multiscale nature of the neural activity produced by NFTsim has the potential to enable the modeling of resulting quantities measurable via various neuroimaging techniques. In this work, we give a comprehensive description of the design and implementation of the software. Due to its modularity and flexibility, NFTsim enables the systematic study of an unlimited number of neural systems with multiple neural populations under a unified framework and allows for direct comparison with analytic and experimental predictions. The code is written in C++ and bundled with Matlab routines for a rapid quantitative analysis and visualization of the outputs. The output of NFTsim is stored in plain text file enabling users to select from a broad range of tools for offline analysis. This software enables a wide and convenient use of powerful physiologically-based neural field approaches to brain modeling. NFTsim is distributed under the Apache 2.0 license.

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.

Sparse Estimation of Resting-State Effective Connectivity From fMRI Cross-Spectra

In functional magnetic resonance imaging (fMRI), functional connectivity is conventionally characterized by correlations between fMRI time series, which are intrinsically undirected measures of connectivity. Yet, some information about the directionality of network connections can nevertheless be extracted from the matrix of pairwise temporal correlations between all considered time series, when expressed in the frequency-domain as a cross-spectral density matrix. Using a sparsity prior, it then becomes possible to determine a unique directed network topology that best explains the observed undirected correlations, without having to rely on temporal precedence relationships that may not be valid in fMRI. Applying this method on simulated data with 100 nodes yielded excellent retrieval of the underlying directed networks under a wide variety of conditions. Importantly, the method did not depend on temporal precedence to establish directionality, thus reducing susceptibility to hemodynamic variability. The computational efficiency of the algorithm was sufficient to enable whole-brain estimations, thus circumventing the problem of missing nodes that otherwise occurs in partial-brain analyses. Applying the method to real resting-state fMRI data acquired with a high temporal resolution, the inferred networks showed good consistency with structural connectivity obtained from diffusion tractography in the same subjects. Interestingly, this agreement could also be seen when considering high-frequency rather than low-frequency connectivity (average correlation: r = 0.26 for f < 0.3 Hz, r = 0.43 for 0.3 < f < 5 Hz). Moreover, this concordance was significantly better (p < 0.05) than for networks obtained with conventional functional connectivity based on correlations (average correlation r = 0.18). The presented methodology thus appears to be well-suited for fMRI, particularly given its lack of explicit dependence on temporal lag structure, and is readily applicable to whole-brain effective connectivity estimation.

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.

The balanced and introspective brain

Transfers of large-scale neural activity into, within and between corticothalamic neural populations and brain hemispheres are analysed using time-integrated transfer functions and state parameters obtained from neural field theory for a variety of arousal states. It is shown that the great majority of activity results from feedbacks within the corticothalamic system, including significant transfer between hemispheres, but only a small minority arises via net input from the external world, with the brain thus in a near-critical, highly introspective state. Notably, the total excitatory and inhibitory influences on cortical neurons are balanced to within a few per cent across arousal states. Strong negative intrahemispheric feedforward exists to the cortex, and even larger interhemispheric positive feedforward, but these are modified by feedback loops to yield near-critical positive overall gain. The results underline the utility of transfer functions for the analysis of brain activity.

Spectral mapping of brain functional connectivity from diffusion imaging

Understanding the relationship between the dynamics of neural processes and the anatomical substrate of the brain is a central question in neuroscience. On the one hand, modern neuroimaging technologies, such as diffusion tensor imaging, can be used to construct structural graphs representing the architecture of white matter streamlines linking cortical and subcortical structures. On the other hand, temporal patterns of neural activity can be used to construct functional graphs representing temporal correlations between brain regions. Although some studies provide evidence that whole-brain functional connectivity is shaped by the underlying anatomy, the observed relationship between function and structure is weak, and the rules by which anatomy constrains brain dynamics remain elusive. In this article, we introduce a methodology to map the functional connectivity of a subject at rest from his or her structural graph. Using our methodology, we are able to systematically account for the role of structural walks in the formation of functional correlations. Furthermore, in our empirical evaluations, we observe that the eigenmodes of the mapped functional connectivity are associated with activity patterns associated with different cognitive systems.

Multi-scale account of the network structure of macaque visual cortex

Cortical network structure has been extensively characterized at the level of local circuits and in terms of long-range connectivity, but seldom in a manner that integrates both of these scales. Furthermore, while the connectivity of cortex is known to be related to its architecture, this knowledge has not been used to derive a comprehensive cortical connectivity map. In this study, we integrate data on cortical architecture and axonal tracing data into a consistent multi-scale framework of the structure of one hemisphere of macaque vision-related cortex. The connectivity model predicts the connection probability between any two neurons based on their types and locations within areas and layers. Our analysis reveals regularities of cortical structure. We confirm that cortical thickness decays with cell density. A gradual reduction in neuron density together with the relative constancy of the volume density of synapses across cortical areas yields denser connectivity in visual areas more remote from sensory inputs and of lower structural differentiation. Further, we find a systematic relation between laminar patterns on source and target sides of cortical projections, extending previous findings from combined anterograde and retrograde tracing experiments. Going beyond the classical schemes, we statistically assign synapses to target neurons based on anatomical reconstructions, which suggests that layer 4 neurons receive substantial feedback input. Our derived connectivity exhibits a community structure that corresponds more closely with known functional groupings than previous connectivity maps and identifies layer-specific directional differences in cortico-cortical pathways. The resulting network can form the basis for studies relating structure to neural dynamics in mammalian cortex at multiple scales.

Cortical geometry as a determinant of brain activity eigenmodes: Neural field analysis

Perturbation analysis of neural field theory is used to derive eigenmodes of neural activity on a cortical hemisphere, which have previously been calculated numerically and found to be close analogs of spherical harmonics, despite heavy cortical folding. The present perturbation method treats cortical folding as a first-order perturbation from a spherical geometry. The first nine spatial eigenmodes on a population-averaged cortical hemisphere are derived and compared with previous numerical solutions. These eigenmodes contribute most to brain activity patterns such as those seen in electroencephalography and functional magnetic resonance imaging. The eigenvalues of these eigenmodes are found to agree with the previous numerical solutions to within their uncertainties. Also in agreement with the previous numerics, all eigenmodes are found to closely resemble spherical harmonics. The first seven eigenmodes exhibit a one-to-one correspondence with their numerical counterparts, with overlaps that are close to unity. The next two eigenmodes overlap the corresponding pair of numerical eigenmodes, having been rotated within the subspace spanned by that pair, likely due to second-order effects. The spatial orientations of the eigenmodes are found to be fixed by gross cortical shape rather than finer-scale cortical properties, which is consistent with the observed intersubject consistency of functional connectivity patterns. However, the eigenvalues depend more sensitively on finer-scale cortical structure, implying that the eigenfrequencies and consequent dynamical properties of functional connectivity depend more strongly on details of individual cortical folding. Overall, these results imply that well-established tools from perturbation theory and spherical harmonic analysis can be used to calculate the main properties and dynamics of low-order brain eigenmodes.

Brain network eigenmodes provide a robust and compact representation of the structural connectome in health and disease

Recent research has demonstrated the use of the structural connectome as a powerful tool to characterize the network architecture of the brain and potentially generate biomarkers for neurologic and psychiatric disorders. In particular, the anatomic embedding of the edges of the cerebral graph have been postulated to elucidate the relative importance of white matter tracts to the overall network connectivity, explaining the varying effects of localized white matter pathology on cognition and behavior. Here, we demonstrate the use of a linear diffusion model to quantify the impact of these perturbations on brain connectivity. We show that the eigenmodes governing the dynamics of this model are strongly conserved between healthy subjects regardless of cortical and sub-cortical parcellations, but show significant, interpretable deviations in improperly developed brains. More specifically, we investigated the effect of agenesis of the corpus callosum (AgCC), one of the most common brain malformations to identify differences in the effect of virtual corpus callosotomies and the neurodevelopmental disorder itself. These findings, including the strong correspondence between regions of highest importance from graph eigenmodes of network diffusion and nexus regions of white matter from edge density imaging, show converging evidence toward understanding the relationship between white matter anatomy and the structural connectome.

While the structural connectome of the brain has emerged as a powerful tool towards understanding the progression of neurologic and psychiatric disorders, links between the anatomy of connections within the brain and the effects of localized white matter pathology on cognition are still an active area of investigation. Here, we propose the use of the diffusion process towards understanding perturbations of brain connectivity. We find that while the dynamics of this process are strongly conserved in healthy subjects, they display significant, interpretable deviations in agenesis of the corpus callosum, one of the most common brain malformations. These findings, including the strong similarity between regions identified to be crucial towards diffusion and nexus regions of white matter from edge density imaging, show converging evidence towards understanding the relationship between white matter anatomy and the structural connectome.

On memories, neural ensembles and mental flexibility

Memories are assumed to be represented by groups of co-activated neurons, called neural ensembles. Describing ensembles is a challenge: complexity of the underlying micro-circuitry is immense. Current approaches use a piecemeal fashion, focusing on single neurons and employing local measures like pairwise correlations. We introduce an alternative approach that identifies ensembles and describes the effective connectivity between them in a holistic fashion. It also links the oscillatory frequencies observed in ensembles with the spatial scales at which activity is expressed. Using unsupervised learning, biophysical modeling and graph theory, we analyze multi-electrode LFPs from frontal cortex during a spatial delayed response task. We find distinct ensembles for different cues and more parsimonious connectivity for cues on the horizontal axis, which may explain the oblique effect in psychophysics. Our approach paves the way for biophysical models with learned parameters that can guide future Brain Computer Interface development.

Model of brain activation predicts the neural collective influence map of the brain

Efficient complex systems have a modular structure, but modularity does not guarantee robustness, because efficiency also requires an ingenious interplay of the interacting modular components. The human brain is the elemental paradigm of an efficient robust modular system interconnected as a network of networks (NoN). Understanding the emergence of robustness in such modular architectures from the interconnections of its parts is a longstanding challenge that has concerned many scientists. Current models of dependencies in NoN inspired by the power grid express interactions among modules with fragile couplings that amplify even small shocks, thus preventing functionality. Therefore, we introduce a model of NoN to shape the pattern of brain activations to form a modular environment that is robust. The model predicts the map of neural collective influencers (NCIs) in the brain, through the optimization of the influence of the minimal set of essential nodes responsible for broadcasting information to the whole-brain NoN. Our results suggest intervention protocols to control brain activity by targeting influential neural nodes predicted by network theory.

Inference of direct and multistep effective connectivities from functional connectivity of the brain and of relationships to cortical geometry

BACKGROUND

The problem of inferring effective brain connectivity from functional connectivity is under active investigation, and connectivity via multistep paths is poorly understood.

NEW METHOD

A method is presented to calculate the direct effective connection matrix (deCM), which embodies direct connection strengths between brain regions, from functional CMs (fCMs) by minimizing the difference between an experimental fCM and one calculated via neural field theory from an ansatz deCM based on an experimental anatomical CM.

RESULTS

The best match between fCMs occurs close to a critical point, consistent with independent published stability estimates. Residual mismatch between fCMs is identified to be largely due to interhemispheric connections that are poorly estimated in an initial ansatz deCM due to experimental limitations; improved ansatzes substantially reduce the mismatch and enable interhemispheric connections to be estimated. Various levels of significant multistep connections are then imaged via the neural field theory (NFT) result that these correspond to powers of the deCM; these are shown to be predictable from geometric distances between regions.

COMPARISON WITH EXISTING METHODS

This method gives insight into direct and multistep effective connectivity from fCMs and relating to physiology and brain geometry. This contrasts with other methods, which progressively adjust connections without an overarching physiologically based framework to deal with multistep or poorly estimated connections.

CONCLUSIONS

deCMs can be usefully estimated using this method and the results enable multistep connections to be investigated systematically.

Time evolution of interhemispheric coupling in a model of focal neocortical epilepsy in mice

Epilepsy is characterized by substantial network rearrangements leading to spontaneous seizures and little is known on how an epileptogenic focus impacts on neural activity in the contralateral hemisphere. Here, we used a model of unilateral epilepsy induced by injection of the synaptic blocker tetanus neurotoxin (TeNT) in the mouse primary visual cortex (V1). Local field potential (LFP) signals were simultaneously recorded from both hemispheres of each mouse in acute phase (peak of toxin action) and chronic condition (completion of TeNT effects). To characterize the neural electrical activities the corresponding LFP signals were analyzed with several methods of time series analysis. For the epileptic mice, the spectral analysis showed that TeNT determines a power redistribution among the different neurophysiological bands in both acute and chronic phases. Using linear and nonlinear interdependence measures in both time and frequency domains, it was found in the acute phase that TeNT injection promotes a reduction of the interhemispheric coupling for high frequencies (12-30 Hz) and small time lag (<20 ms), whereas an increase of the coupling is present for low frequencies (0.5-4 Hz) and long time lag (>40 ms). On the other hand, the chronic period is characterized by a partial or complete recovery of the interhemispheric interdependence level. Granger causality test and symbolic transfer entropy indicate a greater driving influence of the TeNT-injected side on activity in the contralateral hemisphere in the chronic phase. Lastly, based on experimental observations, we built a computational model of LFPs to investigate the role of the ipsilateral inhibition and exicitatory interhemispheric connections in the dampening of the interhemispheric coupling. The time evolution of the interhemispheric coupling in such a relevant model of epilepsy has been addressed here.

Analytical Operations Relate Structural and Functional Connectivity in the Brain

Resting-state large-scale brain models vary in the amount of biological elements they incorporate and in the way they are being tested. One might expect that the more realistic the model is, the closer it should reproduce real functional data. It has been shown, instead, that when linear correlation across long BOLD fMRI time-series is used as a measure for functional connectivity (FC) to compare simulated and real data, a simple model performs just as well, or even better, than more sophisticated ones. The model in question is a simple linear model, which considers the physiological noise that is pervasively present in our brain while it diffuses across the white-matter connections, that is structural connectivity (SC). We deeply investigate this linear model, providing an analytical solution to straightforwardly compute FC from SC without the need of computationally costly simulations of time-series. We provide a few examples how this analytical solution could be used to perform a fast and detailed parameter exploration or to investigate resting-state non-stationarities. Most importantly, by inverting the analytical solution, we propose a method to retrieve information on the anatomical structure directly from functional data. This simple method can be used to complement or guide DTI/DSI and tractography results, especially for a better assessment of inter-hemispheric connections, or to provide an estimate of SC when only functional data are available.

Whole-brain analytic measures of network communication reveal increased structure-function correlation in right temporal lobe epilepsy

The in vivo structure-function relationship is key to understanding brain network reorganization due to pathologies. This relationship is likely to be particularly complex in brain network diseases such as temporal lobe epilepsy, in which disturbed large-scale systems are involved in both transient electrical events and long-lasting functional and structural impairments. Herein, we estimated this relationship by analyzing the correlation between structural connectivity and functional connectivity in terms of analytical network communication parameters. As such, we targeted the gradual topological structure-function reorganization caused by the pathology not only at the whole brain scale but also both in core and peripheral regions of the brain. We acquired diffusion (dMRI) and resting-state fMRI (rsfMRI) data in seven right-lateralized TLE (rTLE) patients and fourteen healthy controls and analyzed the structure-function relationship by using analytical network communication metrics derived from the structural connectome. In rTLE patients, we found a widespread hypercorrelated functional network. Network communication analysis revealed greater unspecific branching of the shortest path (search information) in the structural connectome and a higher global correlation between the structural and functional connectivity for the patient group. We also found evidence for a preserved structural rich-club in the patient group. In sum, global augmentation of structure-function correlation might be linked to a smaller functional repertoire in rTLE patients, while sparing the central core of the brain which may represent a pathway that facilitates the spread of seizures.

Eigenmodes of brain activity: Neural field theory predictions and comparison with experiment

Neural field theory of the corticothalamic system is applied to predict and analyze the activity eigenmodes of the bihemispheric brain, focusing particularly on their spatial structure. The eigenmodes of a single brain hemisphere are found to be close analogs of spherical harmonics, which are the natural modes of the sphere. Instead of multiple eigenvalues being equal, as in the spherical case, cortical folding splits them to have distinct values. Inclusion of interhemispheric connections between homologous regions via the corpus callosum leads to further splitting that depends on symmetry or antisymmetry of activity between brain hemispheres, and the strength and sign of the interhemispheric connections. Symmetry properties of the lowest observed eigenmodes strongly constrain the interhemispheric connectivity strengths and unihemispheric mode spectra, and it is predicted that most spontaneous brain activity will be symmetric between hemispheres, consistent with observations. Comparison with the eigenmodes of an experimental anatomical connectivity matrix confirms these results, permits the relative strengths of intrahemispheric and interhemispheric connectivities to be approximately inferred from their eigenvalues, and lays the foundation for further experimental tests. The results are consistent with brain activity being in corticothalamic eigenmodes, rather than discrete "networks" and open the way to new approaches to brain analysis.

A Mapping Between Structural and Functional Brain Networks

The relationship between structural and functional brain networks is still highly debated. Most previous studies have used a single functional imaging modality to analyze this relationship. In this work, we use multimodal data, from functional MRI, magnetoencephalography, and diffusion tensor imaging, and assume that there exists a mapping between the connectivity matrices of the resting-state functional and structural networks. We investigate this mapping employing group averaged as well as individual data. We indeed find a significantly high goodness of fit level for this structure-function mapping. Our analysis suggests that a functional connection is shaped by all walks up to the diameter in the structural network in both modality cases. When analyzing the inverse mapping, from function to structure, longer walks in the functional network also seem to possess minor influence on the structural connection strengths. Even though similar overall properties for the structure-function mapping are found for different functional modalities, our results indicate that the structure-function relationship is modality dependent.

Estimation of Directed Effective Connectivity from fMRI Functional Connectivity Hints at Asymmetries of Cortical Connectome

The brain exhibits complex spatio-temporal patterns of activity. This phenomenon is governed by an interplay between the internal neural dynamics of cortical areas and their connectivity. Uncovering this complex relationship has raised much interest, both for theory and the interpretation of experimental data (e.g., fMRI recordings) using dynamical models. Here we focus on the so-called inverse problem: the inference of network parameters in a cortical model to reproduce empirically observed activity. Although it has received a lot of interest, recovering directed connectivity for large networks has been rather unsuccessful so far. The present study specifically addresses this point for a noise-diffusion network model. We develop a Lyapunov optimization that iteratively tunes the network connectivity in order to reproduce second-order moments of the node activity, or functional connectivity. We show theoretically and numerically that the use of covariances with both zero and non-zero time shifts is the key to infer directed connectivity. The first main theoretical finding is that an accurate estimation of the underlying network connectivity requires that the time shift for covariances is matched with the time constant of the dynamical system. In addition to the network connectivity, we also adjust the intrinsic noise received by each network node. The framework is applied to experimental fMRI data recorded for subjects at rest. Diffusion-weighted MRI data provide an estimate of anatomical connections, which is incorporated to constrain the cortical model. The empirical covariance structure is reproduced faithfully, especially its temporal component (i.e., time-shifted covariances) in addition to the spatial component that is usually the focus of studies. We find that the cortical interactions, referred to as effective connectivity, in the tuned model are not reciprocal. In particular, hubs are either receptors or feeders: they do not exhibit both strong incoming and outgoing connections. Our results sets a quantitative ground to explore the propagation of activity in the cortex.

The study of interactions between different cortical regions at rest or during a task has considerably developed in the past decades thanks to progress in non-invasive imaging techniques, such as fMRI, EEG and MEG. These techniques have revealed that distant cortical areas exhibit specific correlated activity during the resting state, also called functional connectivity (FC). Moreover, recent studies have highlighted the possible role of white-matter projections between cortical regions in shaping these activity patterns. This structural connectivity (SC) can be estimated using MRI, which measures the probability for two areas to be connected via the density of neural fibers. However, this does not provide the strengths of dynamical interactions. Many methods have thus been developed to estimate the connectivity between neural populations in the cortex that is hypothesized to shape FC. The strengths of these dynamical interactions are called effective connectivity (EC). We use a cortical model that combines information from Diffusion-weighted MRI (dwMRI) and fMRI in order to estimate EC. We demonstrate theoretically that directed C can be inferred using time-shifted covariances. The key point of our method is the use of temporal information from FC at the scale of the whole network. Applying our model on experimental fMRI data at rest, we estimate the asymmetry of intracortical connectivity. Obtaining an accurate EC estimate is essential to analyze its graph properties, such as hubs. In particular, directed connectivity links to the asymmetry between input and output EC strengths of each node, which characterizes feeder and receiver hubs in the cortical network.

Scalability of Asynchronous Networks Is Limited by One-to-One Mapping between Effective Connectivity and Correlations

Network models are routinely downscaled compared to nature in terms of numbers of nodes or edges because of a lack of computational resources, often without explicit mention of the limitations this entails. While reliable methods have long existed to adjust parameters such that the first-order statistics of network dynamics are conserved, here we show that limitations already arise if also second-order statistics are to be maintained. The temporal structure of pairwise averaged correlations in the activity of recurrent networks is determined by the effective population-level connectivity. We first show that in general the converse is also true and explicitly mention degenerate cases when this one-to-one relationship does not hold. The one-to-one correspondence between effective connectivity and the temporal structure of pairwise averaged correlations implies that network scalings should preserve the effective connectivity if pairwise averaged correlations are to be held constant. Changes in effective connectivity can even push a network from a linearly stable to an unstable, oscillatory regime and vice versa. On this basis, we derive conditions for the preservation of both mean population-averaged activities and pairwise averaged correlations under a change in numbers of neurons or synapses in the asynchronous regime typical of cortical networks. We find that mean activities and correlation structure can be maintained by an appropriate scaling of the synaptic weights, but only over a range of numbers of synapses that is limited by the variance of external inputs to the network. Our results therefore show that the reducibility of asynchronous networks is fundamentally limited.

Determination of effective brain connectivity from functional connectivity using propagator-based interferometry and neural field theory with application to the corticothalamic system

It is shown how to compute both direct and total effective connection matrices (deCMs and teCMs), which embody the strengths of neural connections between regions, from correlation-based functional CMs using propagator-based interferometry, a method that stems from geophysics and acoustics, coupled with the recent identification of deCMs and teCMs with bare and dressed propagators, respectively. The approach incorporates excitatory and inhibitory connections, multiple structures and populations, and measurement effects. The propagator is found for a generalized scalar wave equation derived from neural field theory, and expressed in terms of neural activity correlations and covariances, and wave damping rates. It is then related to correlation matrices that are commonly used to express functional and effective connectivities in the brain. The results are illustrated in analytically tractable test cases.