Spectral graph theory of brain oscillations

Structure-function coupling in macroscale human brain networks

Precisely how the anatomical structure of the brain gives rise to a repertoire of complex functions remains incompletely understood. A promising manifestation of this mapping from structure to function is the dependency of the functional activity of a brain region on the underlying white matter architecture. Here, we review the literature examining the macroscale coupling between structural and functional connectivity, and we establish how this structure-function coupling (SFC) can provide more information about the underlying workings of the brain than either feature alone. We begin by defining SFC and describing the computational methods used to quantify it. We then review empirical studies that examine the heterogeneous expression of SFC across different brain regions, among individuals, in the context of the cognitive task being performed, and over time, as well as its role in fostering flexible cognition. Last, we investigate how the coupling between structure and function is affected in neurological and psychiatric conditions, and we report how aberrant SFC is associated with disease duration and disease-specific cognitive impairment. By elucidating how the dynamic relationship between the structure and function of the brain is altered in the presence of neurological and psychiatric conditions, we aim to not only further our understanding of their aetiology but also establish SFC as a new and sensitive marker of disease symptomatology and cognitive performance. Overall, this Review collates the current knowledge regarding the regional interdependency between the macroscale structure and function of the human brain in both neurotypical and neuroatypical individuals.

Simulation-based Inference of Developmental EEG Maturation with the Spectral Graph Model

The spectral content of macroscopic neural activity evolves throughout development, yet how this maturation relates to underlying brain network formation and dynamics remains unknown. Here, we assess the developmental maturation of electroencephalogram spectra via Bayesian model inversion of the spectral graph model, a parsimonious whole-brain model of spatiospectral neural activity derived from linearized neural field models coupled by the structural connectome. Simulation-based inference was used to estimate age-varying spectral graph model parameter posterior distributions from electroencephalogram spectra spanning the developmental period. This model-fitting approach accurately captures observed developmental electroencephalogram spectral maturation via a neurobiologically consistent progression of key neural parameters: long-range coupling, axonal conduction speed, and excitatory:inhibitory balance. These results suggest that the spectral maturation of macroscopic neural activity observed during typical development is supported by age-dependent functional adaptations in localized neural dynamics and their long-range coupling across the macroscopic structural network.

Spectral graph model for fMRI: a biophysical, connectivity-based generative model for the analysis of frequency-resolved resting state fMRI

Resting state functional MRI (rs-fMRI) is a popular and widely used technique to explore the brain's functional organization and to examine if it is altered in neurological or mental disorders. The most common approach for its analysis targets the measurement of the synchronized fluctuations between brain regions, characterized as functional connectivity (FC), typically relying on pairwise correlations in activity across different brain regions. While hugely successful in exploring state- and disease-dependent network alterations, these statistical graph theory tools suffer from two key limitations. First, they discard useful information about the rich frequency content of the fMRI signal. The rich spectral information now achievable from advances in fast multiband acquisitions is consequently being under-utilized. Second, the analyzed FCs are phenomenological without a direct neurobiological underpinning in the underlying structures and processes in the brain. There does not currently exist a complete generative model framework for whole brain resting fMRI that is informed by its underlying biological basis in the structural connectome. Here we propose that a different approach can solve both challenges at once: the use of an appropriately realistic yet parsimonious biophysical signal generation model followed by graph spectral (i.e. eigen) decomposition. We call this model a Spectral Graph Model (SGM) for fMRI, using which we can not only quantify the structure-function relationship in individual subjects, but also condense the variable and individual-specific repertoire of fMRI signal's spectral and spatial features into a small number of biophysically-interpretable parameters. We expect this model-based inference of rs-fMRI that seamlessly integrates with structure can be used to examine state and trait characteristics of structure-function relations in a variety of brain disorders.

Impaired long-range excitatory time scale predicts abnormal neural oscillations and cognitive deficits in Alzheimer's disease

BACKGROUND

Alzheimer's disease (AD) is the most common form of dementia, progressively impairing cognitive abilities. While neuroimaging studies have revealed functional abnormalities in AD, how these relate to aberrant neuronal circuit mechanisms remains unclear. Using magnetoencephalography imaging we documented abnormal local neural synchrony patterns in patients with AD. To identify global abnormal biophysical mechanisms underlying the spatial and spectral electrophysiological patterns in AD, we estimated the parameters of a biophysical spectral graph model (SGM).

METHODS

SGM is an analytic neural mass model that describes how long-range fiber projections in the brain mediate the excitatory and inhibitory activity of local neuronal subpopulations. Unlike other coupled neuronal mass models, the SGM is linear, available in closed-form, and parameterized by a small set of biophysical interpretable global parameters. This facilitates their rapid and unambiguous inference which we performed here on a well-characterized clinical population of patients with AD (N = 88, age = 62.73 +/- 8.64 years) and a cohort of age-matched controls (N = 88, age = 65.07 +/- 9.92 years).

RESULTS

Patients with AD showed significantly elevated long-range excitatory neuronal time scales, local excitatory neuronal time scales and local inhibitory neural synaptic strength. The long-range excitatory time scale had a larger effect size, compared to local excitatory time scale and inhibitory synaptic strength and contributed highest for the accurate classification of patients with AD from controls. Furthermore, increased long-range time scale was associated with greater deficits in global cognition.

CONCLUSIONS

These results demonstrate that long-range excitatory time scale of neuronal activity, despite being a global measure, is a key determinant in the local spectral signatures and cognition in the human brain, and how it might be a parsimonious factor underlying altered neuronal activity in AD. Our findings provide new insights into mechanistic links between abnormal local spectral signatures and global connectivity measures in AD.

Cortical Synchrony and Information Flow during Transition from Wakefulness to Light Non-Rapid Eye Movement Sleep

Sleep is a highly stereotyped phenomenon, requiring robust spatiotemporal coordination of neural activity. Understanding how the brain coordinates neural activity with sleep onset can provide insights into the physiological functions subserved by sleep and the pathologic phenomena associated with sleep onset. We quantified whole-brain network changes in synchrony and information flow during the transition from wakefulness to light non-rapid eye movement (NREM) sleep, using MEG imaging in a convenient sample of 14 healthy human participants (11 female; mean 63.4 years [SD 11.8 years]). We furthermore performed computational modeling to infer excitatory and inhibitory properties of local neural activity. The transition from wakefulness to light NREM was identified to be encoded in spatially and temporally specific patterns of long-range synchrony. Within the delta band, there was a global increase in connectivity from wakefulness to light NREM, which was highest in frontoparietal regions. Within the theta band, there was an increase in connectivity in fronto-parieto-occipital regions and a decrease in temporal regions from wakefulness to Stage 1 sleep. Patterns of information flow revealed that mesial frontal regions receive hierarchically organized inputs from broad cortical regions upon sleep onset, including direct inflow from occipital regions and indirect inflow via parieto-temporal regions within the delta frequency band. Finally, biophysical neural mass modeling demonstrated changes in the anterior-to-posterior distribution of cortical excitation-to-inhibition with increased excitation-to-inhibition model parameters in anterior regions in light NREM compared with wakefulness. Together, these findings uncover whole-brain corticocortical structure and the orchestration of local and long-range, frequency-specific cortical interactions in the sleep-wake transition.SIGNIFICANCE STATEMENT Our work uncovers spatiotemporal cortical structure of neural synchrony and information flow upon the transition from wakefulness to light non-rapid eye movement sleep. Mesial frontal regions were identified to receive hierarchically organized inputs from broad cortical regions, including both direct inputs from occipital regions and indirect inputs via the parieto-temporal regions within the delta frequency range. Biophysical neural mass modeling revealed a spatially heterogeneous, anterior-posterior distribution of cortical excitation-to-inhibition. Our findings shed light on the orchestration of local and long-range cortical neural structure that is fundamental to sleep onset, and support an emerging view of cortically driven regulation of sleep homeostasis.

Impaired long-range excitatory time scale predicts abnormal neural oscillations and cognitive deficits in Alzheimer's disease

Alzheimer's disease (AD) is the most common form of dementia, progressively impairing memory and cognition. While neuroimaging studies have revealed functional abnormalities in AD, how these relate to aberrant neuronal circuit mechanisms remains unclear. Using magnetoencephalography imaging we documented abnormal local neural synchrony patterns in patients with AD. To identify abnormal biophysical mechanisms underlying these abnormal electrophysiological patterns, we estimated the parameters of a spectral graph-theory model (SGM). SGM is an analytic model that describes how long-range fiber projections in the brain mediate the excitatory and inhibitory activity of local neuronal subpopulations. The long-range excitatory time scale was associated with greater deficits in global cognition and was able to distinguish AD patients from controls with high accuracy. These results demonstrate that long-range excitatory time scale of neuronal activity, despite being a global measure, is a key determinant in the spatiospectral signatures and cognition in AD.

Impaired long-range excitatory time scale predicts abnormal neural oscillations and cognitive deficits in Alzheimer's disease

Alzheimer's disease (AD) is the most common form of dementia, progressively impairing memory and cognition. While neuroimaging studies have revealed functional abnormalities in AD, how these relate to aberrant neuronal circuit mechanisms remains unclear. Using magnetoencephalography imaging we documented abnormal local neural synchrony patterns in patients with AD. To identify abnormal biophysical mechanisms underlying these abnormal electrophysiological patterns, we estimated the parameters of a spectral graph-theory model (SGM). SGM is an analytic model that describes how long-range fiber projections in the brain mediate the excitatory and inhibitory activity of local neuronal subpopulations. The long-range excitatory time scale was associated with greater deficits in global cognition and was able to distinguish AD patients from controls with high accuracy. These results demonstrate that long-range excitatory time scale of neuronal activity, despite being a global measure, is a key determinant in the spatiospectral signatures and cognition in AD.

Enhanced brain structure-function tethering in transmodal cortex revealed by high-frequency eigenmodes

While the link between brain structure and function remains an ongoing challenge, the prevailing hypothesis is that the structure-function relationship may itself be gradually decoupling from unimodal to transmodal cortex. However, this hypothesis is constrained by the underlying models which may neglect requisite information. Here we relate structural and functional connectivity derived from diffusion and functional MRI through orthogonal eigenmodes governing frequency-specific diffusion patterns. We find that low-frequency eigenmodes contribute little to functional interactions in transmodal cortex, resulting in divergent structure-function relationships. Conversely, high-frequency eigenmodes predominantly support neuronal coactivation patterns in these areas, inducing structure-function convergence along a unimodal-transmodal hierarchy. High-frequency information, although weak and scattered, could enhance the structure-function tethering, especially in transmodal association cortices. Our findings suggest that the structure-function decoupling may not be an intrinsic property of brain organization, but can be narrowed through multiplexed and regionally specialized spatiotemporal propagation regimes.

The coupling between the spatial and temporal scales of neural processes revealed by a joint time-vertex connectome spectral analysis

Brain oscillations are produced by the coordinated activity of large groups of neurons and different rhythms are thought to reflect different modes of information processing. These modes, in turn, are known to occur at different spatial scales. Nevertheless, how these rhythms support different spatial modes of information processing at the brain scale is not yet fully understood. Here we use "Joint Time-Vertex Spectral Analysis" to characterize the joint spectral content of brain activity both in time (temporal frequencies) and in space over the connectivity graph (spatial connectome harmonics). This method allows us to characterize the relationship between spatially localized or distributed neural processes on one side and their respective temporal frequency bands in source-reconstructed M/EEG signals. We explore this approach on two different datasets, an auditory steady-state response (ASSR) and a visual grating task. Our results suggest that different information processing mechanisms are carried out at different frequency bands: while spatially distributed activity (which may also be interpreted as integration) specifically occurs at low temporal frequencies (alpha and theta) and low graph spatial frequencies, localized electrical activity (i.e., segregation) is observed at high temporal frequencies (high and low gamma) over restricted high spatial graph frequencies. Crucially, the estimated contribution of the distributed and localized neural activity predicts performance in a behavioral task, demonstrating the neurophysiological relevance of the joint time-vertex spectral representation.

Bayesian inference of a spectral graph model for brain oscillations

The relationship between brain functional connectivity and structural connectivity has caught extensive attention of the neuroscience community, commonly inferred using mathematical modeling. Among many modeling approaches, spectral graph model (SGM) is distinctive as it has a closed-form solution of the wide-band frequency spectra of brain oscillations, requiring only global biophysically interpretable parameters. While SGM is parsimonious in parameters, the determination of SGM parameters is non-trivial. Prior works on SGM determine the parameters through a computational intensive annealing algorithm, which only provides a point estimate with no confidence intervals for parameter estimates. To fill this gap, we incorporate the simulation-based inference (SBI) algorithm and develop a Bayesian procedure for inferring the posterior distribution of the SGM parameters. Furthermore, using SBI dramatically reduces the computational burden for inferring the SGM parameters. We evaluate the proposed SBI-SGM framework on the resting-state magnetoencephalography recordings from healthy subjects and show that the proposed procedure has similar performance to the annealing algorithm in recovering power spectra and the spatial distribution of the alpha frequency band. In addition, we also analyze the correlations among the parameters and their uncertainty with the posterior distribution which cannot be done with annealing inference. These analyses provide a richer understanding of the interactions among biophysical parameters of the SGM. In general, the use of simulation-based Bayesian inference enables robust and efficient computations of generative model parameter uncertainties and may pave the way for the use of generative models in clinical translation applications.

Comparing individual and group-level simulated neurophysiological brain connectivity using the Jansen and Rit neural mass model

Computational models are often used to assess how functional connectivity (FC) patterns emerge from neuronal population dynamics and anatomical brain connections. It remains unclear whether the commonly used group-averaged data can predict individual FC patterns. The Jansen and Rit neural mass model was employed, where masses were coupled using individual structural connectivity (SC). Simulated FC was correlated to individual magnetoencephalography-derived empirical FC. FC was estimated using phase-based (phase lag index (PLI), phase locking value (PLV)), and amplitude-based (amplitude envelope correlation (AEC)) metrics to analyze their goodness of fit for individual predictions. Individual FC predictions were compared against group-averaged FC predictions, and we tested whether SC of a different participant could equally well predict participants' FC patterns. The AEC provided a better match between individually simulated and empirical FC than phase-based metrics. Correlations between simulated and empirical FC were higher using individual SC compared to group-averaged SC. Using SC from other participants resulted in similar correlations between simulated and empirical FC compared to using participants' own SC. This work underlines the added value of FC simulations using individual instead of group-averaged SC for this particular computational model and could aid in a better understanding of mechanisms underlying individual functional network trajectories.

Multi-modal and multi-model interrogation of large-scale functional brain networks

Existing whole-brain models are generally tailored to the modelling of a particular data modality (e.g., fMRI or MEG/EEG). We propose that despite the differing aspects of neural activity each modality captures, they originate from shared network dynamics. Building on the universal principles of self-organising delay-coupled nonlinear systems, we aim to link distinct features of brain activity - captured across modalities - to the dynamics unfolding on a macroscopic structural connectome. To jointly predict connectivity, spatiotemporal and transient features of distinct signal modalities, we consider two large-scale models - the Stuart Landau and Wilson and Cowan models - which generate short-lived 40 Hz oscillations with varying levels of realism. To this end, we measure features of functional connectivity and metastable oscillatory modes (MOMs) in fMRI and MEG signals - and compare them against simulated data. We show that both models can represent MEG functional connectivity (FC), functional connectivity dynamics (FCD) and generate MOMs to a comparable degree. This is achieved by adjusting the global coupling and mean conduction time delay and, in the WC model, through the inclusion of balance between excitation and inhibition. For both models, the omission of delays dramatically decreased the performance. For fMRI, the SL model performed worse for FCD and MOMs, highlighting the importance of balanced dynamics for the emergence of spatiotemporal and transient patterns of ultra-slow dynamics. Notably, optimal working points varied across modalities and no model was able to achieve a correlation with empirical FC higher than 0.4 across modalities for the same set of parameters. Nonetheless, both displayed the emergence of FC patterns that extended beyond the constraints of the anatomical structure. Finally, we show that both models can generate MOMs with empirical-like properties such as size (number of brain regions engaging in a mode) and duration (continuous time interval during which a mode appears). Our results demonstrate the emergence of static and dynamic properties of neural activity at different timescales from networks of delay-coupled oscillators at 40 Hz. Given the higher dependence of simulated FC on the underlying structural connectivity, we suggest that mesoscale heterogeneities in neural circuitry may be critical for the emergence of parallel cross-modal functional networks and should be accounted for in future modelling endeavours.

A joint subspace mapping between structural and functional brain connectomes

Understanding the connection between the brain's structural connectivity and its functional connectivity is of immense interest in computational neuroscience. Although some studies have suggested that whole brain functional connectivity is shaped by the underlying structure, the rule by which anatomy constraints brain dynamics remains an open question. In this work, we introduce a computational framework that identifies a joint subspace of eigenmodes for both functional and structural connectomes. We found that a small number of those eigenmodes are sufficient to reconstruct functional connectivity from the structural connectome, thus serving as low-dimensional basis function set. We then develop an algorithm that can estimate the functional eigen spectrum in this joint space from the structural eigen spectrum. By concurrently estimating the joint eigenmodes and the functional eigen spectrum, we can reconstruct a given subject's functional connectivity from their structural connectome. We perform elaborate experiments and demonstrate that the proposed algorithm for estimating functional connectivity from the structural connectome using joint space eigenmodes gives competitive performance as compared to the existing benchmark methods with better interpretability.

Bayesian Inference of a Spectral Graph Model for Brain Oscillations

The relationship between brain functional connectivity and structural connectivity has caught extensive attention of the neuroscience community, commonly inferred using mathematical modeling. Among many modeling approaches, spectral graph model (SGM) is distinctive as it has a closed-form solution of the wide-band frequency spectra of brain oscillations, requiring only global biophysically interpretable parameters. While SGM is parsimonious in parameters, the determination of SGM parameters is non-trivial. Prior works on SGM determine the parameters through a computational intensive annealing algorithm, which only provides a point estimate with no confidence intervals for parameter estimates. To fill this gap, we incorporate the simulation-based inference (SBI) algorithm and develop a Bayesian procedure for inferring the posterior distribution of the SGM parameters. Furthermore, using SBI dramatically reduces the computational burden for inferring the SGM parameters. We evaluate the proposed SBI-SGM framework on the resting-state magnetoencephalography recordings from healthy subjects and show that the proposed procedure has similar performance to the annealing algorithm in recovering power spectra and the spatial distribution of the alpha frequency band. In addition, we also analyze the correlations among the parameters and their uncertainty with the posterior distribution which can not be done with annealing inference. These analyses provide a richer understanding of the interactions among biophysical parameters of the SGM. In general, the use of simulation-based Bayesian inference enables robust and efficient computations of generative model parameter uncertainties and may pave the way for the use of generative models in clinical translation applications.

Bayesian Algorithms for Joint Estimation of Brain Activity and Noise in Electromagnetic Imaging

Simultaneously estimating brain source activity and noise has long been a challenging task in electromagnetic brain imaging using magneto- and electroencephalography. The problem is challenging not only in terms of solving the NP-hard inverse problem of reconstructing unknown brain activity across thousands of voxels from a limited number of sensors, but also for the need to simultaneously estimate the noise and interference. We present a generative model with an augmented leadfield matrix to simultaneously estimate brain source activity and sensor noise statistics in electromagnetic brain imaging (EBI). We then derive three Bayesian inference algorithms for this generative model (expectation-maximization (EBI-EM), convex bounding (EBI-Convex) and fixed-point (EBI-Mackay)) to simultaneously estimate the hyperparameters of the prior distribution for brain source activity and sensor noise. A comprehensive performance evaluation for these three algorithms is performed. Simulations consistently show that the performance of EBI-Convex and EBI-Mackay updates is superior to that of EBI-EM. In contrast to the EBI-EM algorithm, both EBI-Convex and EBI-Mackay updates are quite robust to initialization, and are computationally efficient with fast convergence in the presence of both Gaussian and real brain noise. We also demonstrate that EBI-Convex and EBI-Mackay update algorithms can reconstruct complex brain activity with only a few trials of sensor data, and for resting-state data, achieving significant improvement in source reconstruction and noise learning for electromagnetic brain imaging.

Distributed harmonic patterns of structure-function dependence orchestrate human consciousness

A central question in neuroscience is how consciousness arises from the dynamic interplay of brain structure and function. Here we decompose functional MRI signals from pathological and pharmacologically-induced perturbations of consciousness into distributed patterns of structure-function dependence across scales: the harmonic modes of the human structural connectome. We show that structure-function coupling is a generalisable indicator of consciousness that is under bi-directional neuromodulatory control. We find increased structure-function coupling across scales during loss of consciousness, whether due to anaesthesia or brain injury, capable of discriminating between behaviourally indistinguishable sub-categories of brain-injured patients, tracking the presence of covert consciousness. The opposite harmonic signature characterises the altered state induced by LSD or ketamine, reflecting psychedelic-induced decoupling of brain function from structure and correlating with physiological and subjective scores. Overall, connectome harmonic decomposition reveals how neuromodulation and the network architecture of the human connectome jointly shape consciousness and distributed functional activation across scales.

Structure-function models of temporal, spatial, and spectral characteristics of non-invasive whole brain functional imaging

We review recent advances in using mathematical models of the relationship between the brain structure and function that capture features of brain dynamics. We argue the need for models that can jointly capture temporal, spatial, and spectral features of brain functional activity. We present recent work on spectral graph theory based models that can accurately capture spectral as well as spatial patterns across multiple frequencies in MEG reconstructions.

Predicting time-resolved electrophysiological brain networks from structural eigenmodes

How temporal modulations in functional interactions are shaped by the underlying anatomical connections remains an open question. Here, we analyse the role of structural eigenmodes, in the formation and dissolution of temporally evolving functional brain networks using resting-state magnetoencephalography and diffusion magnetic resonance imaging data at the individual subject level. Our results show that even at short timescales, phase and amplitude connectivity can partly be expressed by structural eigenmodes, but hardly by direct structural connections. Albeit a stronger relationship was found between structural eigenmodes and time-resolved amplitude connectivity. Time-resolved connectivity for both phase and amplitude was mostly characterised by a stationary process, superimposed with very brief periods that showed deviations from this stationary process. For these brief periods, dynamic network states were extracted that showed different expressions of eigenmodes. Furthermore, the eigenmode expression was related to overall cognitive performance and co-occurred with fluctuations in community structure of functional networks. These results implicate that ongoing time-resolved resting-state networks, even at short timescales, can to some extent be understood in terms of activation and deactivation of structural eigenmodes and that these eigenmodes play a role in the dynamic integration and segregation of information across the cortex, subserving cognitive functions.

Altered excitatory and inhibitory neuronal subpopulation parameters are distinctly associated with tau and amyloid in Alzheimer's disease

Background

Neuronal- and circuit-level abnormalities of excitation and inhibition are shown to be associated with tau and amyloid-beta (Aβ) in preclinical models of Alzheimer's disease (AD). These relationships remain poorly understood in patients with AD.

Methods

Using empirical spectra from magnetoencephalography and computational modeling (neural mass model), we examined excitatory and inhibitory parameters of neuronal subpopulations and investigated their specific associations to regional tau and Aβ, measured by positron emission tomography, in patients with AD.

Results

Patients with AD showed abnormal excitatory and inhibitory time-constants and neural gains compared to age-matched controls. Increased excitatory time-constants distinctly correlated with higher tau depositions while increased inhibitory time-constants distinctly correlated with higher Aβ depositions.

Conclusions

Our results provide critical insights about potential mechanistic links between abnormal neural oscillations and cellular correlates of impaired excitatory and inhibitory synaptic functions associated with tau and Aβ in patients with AD.

Funding

This study was supported by the National Institutes of Health grants: K08AG058749 (KGR), F32AG050434-01A1 (KGR), K23 AG038357 (KAV), P50 AG023501, P01 AG19724 (BLM), P50-AG023501 (BLM and GDR), R01 AG045611 (GDR); AG034570, AG062542 (WJ); NS100440 (SSN), DC176960 (SSN), DC017091 (SSN), AG062196 (SSN); a grant from John Douglas French Alzheimer's Foundation (KAV); grants from Larry L. Hillblom Foundation: 2015-A-034-FEL (KGR), 2019-A-013-SUP (KGR); grants from the Alzheimer's Association: AARG-21-849773 (KGR); PCTRB-13-288476 (KAV), and made possible by Part the CloudTM (ETAC-09-133596); a grant from Tau Consortium (GDR and WJJ), and a gift from the S. D. Bechtel Jr. Foundation.

Alpha blocking and 1/fβ spectral scaling in resting EEG can be accounted for by a sum of damped alpha band oscillatory processes

The dynamical and physiological basis of alpha band activity and 1/f^β noise in the EEG are the subject of continued speculation. Here we conjecture, on the basis of empirical data analysis, that both of these features may be economically accounted for through a single process if the resting EEG is conceived of being the sum of multiple stochastically perturbed alpha band damped linear oscillators with a distribution of dampings (relaxation rates). The modulation of alpha-band and 1/f^β noise activity by changes in damping is explored in eyes closed (EC) and eyes open (EO) resting state EEG. We aim to estimate the distribution of dampings by solving an inverse problem applied to EEG power spectra. The characteristics of the damping distribution are examined across subjects, sensors and recording condition (EC/EO). We find that there are robust changes in the damping distribution between EC and EO recording conditions across participants. The estimated damping distributions are found to be predominantly bimodal, with the number and position of the modes related to the sharpness of the alpha resonance and the scaling (β) of the power spectrum (1/f^β). The results suggest that there exists an intimate relationship between resting state alpha activity and 1/f^β noise with changes in both governed by changes to the damping of the underlying alpha oscillatory processes. In particular, alpha-blocking is observed to be the result of the most weakly damped distribution mode becoming more heavily damped. The results suggest a novel way of characterizing resting EEG power spectra and provides new insight into the central role that damped alpha-band activity may play in characterising the spatio-temporal features of resting state EEG.

The resting human electroencephalogram (EEG) exhibits two dominant spectral features: the alpha rhythm (8–13 Hz) and its associated attenuation between eyes-closed and eyes-open resting state (alpha blocking), and the 1/f^β scaling of the power spectrum. While these phenomena are well studied a thorough understanding of their respective generative processes remains elusive. By employing a theoretical approach that follows from neural population models of EEG we demonstrate that it is possible to economically account for both of these phenomena using a singular mechanistic framework: resting EEG is assumed to arise from the summed activity of multiple uncorrelated, stochastically driven, damped alpha band linear oscillatory processes having a distribution of relaxation rates or dampings. By numerically estimating these damping distributions from eyes-closed and eyes-open EEG data, in a total of 136 participants, it is found that such damping distributions are predominantly bimodal in shape. The most weakly damped mode is found to account for alpha band power, with alpha blocking being driven by an increase in the damping of this weakly damped mode, whereas the second, and more heavily damped mode, is able to explain 1/f^β scaling present in the resting state EEG spectra.

Predicting Functional Connectivity From Observed and Latent Structural Connectivity via Eigenvalue Mapping

Understanding how complex dynamic activity propagates over a static structural network is an overarching question in the field of neuroscience. Previous work has demonstrated that linear graph-theoretic models perform as well as non-linear neural simulations in predicting functional connectivity with the added benefits of low dimensionality and a closed-form solution which make them far less computationally expensive. Here we show a simple model relating the eigenvalues of the structural connectivity and functional networks using the Gamma function, producing a reliable prediction of functional connectivity with a single model parameter. We also investigate the impact of local activity diffusion and long-range interhemispheric connectivity on the structure-function model and show an improvement in functional connectivity prediction when accounting for such latent variables which are often excluded from traditional diffusion tensor imaging (DTI) methods.

Spectral graph theory of brain oscillations - revisited and improved

Mathematical modeling of the relationship between the functional activity and the structural wiring of the brain has largely been undertaken using non-linear and biophysically detailed mathematical models with regionally varying parameters. While this approach provides us a rich repertoire of multistable dynamics that can be displayed by the brain, it is computationally demanding. Moreover, although neuronal dynamics at the microscopic level are nonlinear and chaotic, it is unclear if such detailed nonlinear models are required to capture the emergent meso-(regional population ensemble) and macro-scale (whole brain) behavior, which is largely deterministic and reproducible across individuals. Indeed, recent modeling effort based on spectral graph theory has shown that an analytical model without regionally varying parameters and without multistable dynamics can capture the empirical magnetoencephalography frequency spectra and the spatial patterns of the alpha and beta frequency bands accurately.

In this work, we demonstrate an improved hierarchical, linearized, and analytic spectral graph theory-based model that can capture the frequency spectra obtained from magnetoencephalography recordings of resting healthy subjects. We reformulated the spectral graph theory model in line with classical neural mass models, therefore providing more biologically interpretable parameters, especially at the local scale. We demonstrated that this model performs better than the original model when comparing the spectral correlation of modeled frequency spectra and that obtained from the magnetoencephalography recordings. This model also performs equally well in predicting the spatial patterns of the empirical alpha and beta frequency bands.

The connectome spectrum as a canonical basis for a sparse representation of fast brain activity

The functional organization of neural processes is constrained by the brain's intrinsic structural connectivity, i.e., the connectome. Here, we explore how structural connectivity can improve the representation of brain activity signals and their dynamics. Using a multi-modal imaging dataset (electroencephalography, structural MRI, and diffusion MRI), we represent electrical brain activity at the cortical surface as a time-varying composition of harmonic modes of structural connectivity. These harmonic modes are known as connectome harmonics. Here we describe brain activity signal as a time-varying combination of connectome harmonics. We term this description as the connectome spectrum of the signal. We found that: first, the brain activity signal is represented more compactly by the connectome spectrum than by the traditional area-based representation; second, the connectome spectrum characterizes fast brain dynamics in terms of signal broadcasting profile, revealing different temporal regimes of integration and segregation that are consistent across participants. And last, the connectome spectrum characterizes fast brain dynamics with fewer degrees of freedom than area-based signal representations. Specifically, we show that a smaller number of dimensions capture the differences between low-level and high-level visual processing in the connectome spectrum. Also, we demonstrate that connectome harmonics capture more sensitively the topological properties of brain activity. In summary, this work provides statistical, functional, and topological evidence indicating that the description of brain activity in terms of structural connectivity fosters a more comprehensive understanding of large-scale dynamic neural functioning.

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.

Functional harmonics reveal multi-dimensional basis functions underlying cortical organization

The human brain consists of specialized areas that flexibly interact to form a multitude of functional networks. Complementary to this notion of modular organization, brain function has been shown to vary along a smooth continuum across the whole cortex. We demonstrate a mathematical framework that accounts for both of these perspectives: harmonic modes. We calculate the harmonic modes of the brain's functional connectivity graph, called "functional harmonics," revealing a multi-dimensional, frequency-ordered set of basis functions. Functional harmonics link characteristics of cortical organization across several spatial scales, capturing aspects of intra-areal organizational features (retinotopy, somatotopy), delineating brain areas, and explaining macroscopic functional networks as well as global cortical gradients. Furthermore, we show how the activity patterns elicited by seven different tasks are reconstructed from a very small subset of functional harmonics. Our results suggest that the principle of harmonicity, ubiquitous in nature, also underlies functional cortical organization in the human brain.

Improved neuropathological identification of traumatic brain injury through quantitative neuroimaging and neural network analyses: Some practical approaches for the neurorehabilitation clinician

BACKGROUND

Quantitative neuroimaging analyses have the potential to provide additional information about the neuropathology of traumatic brain injury (TBI) that more thoroughly informs the neurorehabilitation clinician.

OBJECTIVE

Quantitative neuroimaging is typically not covered in the standard radiological report, but often can be extracted via post-processing of clinical neuroimaging studies, provided that the proper volume acquisition sequences were originally obtained.

METHODS

Research and commercially available quantitative neuroimaging methods provide region of interest (ROI) quantification metrics, lesion burden volumetrics and cortical thickness measures, degree of focal encephalomalacia, white matter (WM) abnormalities and residual hemorrhagic pathology. If present, diffusion tensor imaging (DTI) provides a variety of techniques that aid in evaluating WM integrity. Using quantitatively identified structural and ROI neuropathological changes are most informative when done from a neural network approach.

RESULTS

Viewing quantitatively identifiable damage from a neural network perspective provides the neurorehabilitation clinician with an additional tool for linking brain pathology to understand symptoms, problems and deficits as well as aid neuropsychological test interpretation. All of these analyses can be displayed in graphic form, including3-D image analysis. A case study approach is used to demonstrate the utility of quantitative neuroimaging and network analyses in TBI.

CONCLUSIONS

Quantitative neuroimaging may provide additional useful information for the neurorehabilitation clinician.

Emergence of canonical functional networks from the structural connectome

How do functional brain networks emerge from the underlying wiring of the brain? We examine how resting-state functional activation patterns emerge from the underlying connectivity and length of white matter fibers that constitute its “structural connectome”. By introducing realistic signal transmission delays along fiber projections, we obtain a complex-valued graph Laplacian matrix that depends on two parameters: coupling strength and oscillation frequency. This complex Laplacian admits a complex-valued eigen-basis in the frequency domain that is highly tunable and capable of reproducing the spatial patterns of canonical functional networks without requiring any detailed neural activity modeling. Specific canonical functional networks can be predicted using linear superposition of small subsets of complex eigenmodes. Using a novel parameter inference procedure we show that the complex Laplacian outperforms the real-valued Laplacian in predicting functional networks. The complex Laplacian eigenmodes therefore constitute a tunable yet parsimonious substrate on which a rich repertoire of realistic functional patterns can emerge. Although brain activity is governed by highly complex nonlinear processes and dense connections, our work suggests that simple extensions of linear models to the complex domain effectively approximate rich macroscopic spatial patterns observable on BOLD fMRI.

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.

Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome

Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed "connectome harmonics", have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI). As an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome graph constructed from diffusion tensor imaging (DTI) and structural MRI data. We show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of resting-state fMRI data, and predict its functional connectivity, with a high level of detail. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.

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.

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. .