Complexity versus modularity and heterogeneity in oscillatory networks: combining segregation and integration in neural systems

A Quantitative Data-Driven Analysis Framework for Resting-State Functional Magnetic Resonance Imaging: A Study of the Impact of Adult Age

The objective of this study is to introduce a new quantitative data-driven analysis (QDA) framework for the analysis of resting-state fMRI (R-fMRI) and use it to investigate the effect of adult age on resting-state functional connectivity (RFC). Whole-brain R-fMRI measurements were conducted on a 3T clinical MRI scanner in 227 healthy adult volunteers (N = 227, aged 18-76 years old, male/female = 99/128). With the proposed QDA framework we derived two types of voxel-wise RFC metrics: the connectivity strength index and connectivity density index utilizing the convolutions of the cross-correlation histogram with different kernels. Furthermore, we assessed the negative and positive portions of these metrics separately. With the QDA framework we found age-related declines of RFC metrics in the superior and middle frontal gyri, posterior cingulate cortex (PCC), right insula and inferior parietal lobule of the default mode network (DMN), which resembles previously reported results using other types of RFC data processing methods. Importantly, our new findings complement previously undocumented results in the following aspects: (1) the PCC and right insula are anti-correlated and tend to manifest simultaneously declines of both the negative and positive connectivity strength with subjects' age; (2) separate assessment of the negative and positive RFC metrics provides enhanced sensitivity to the aging effect; and (3) the sensorimotor network depicts enhanced negative connectivity strength with the adult age. The proposed QDA framework can produce threshold-free and voxel-wise RFC metrics from R-fMRI data. The detected adult age effect is largely consistent with previously reported studies using different R-fMRI analysis approaches. Moreover, the separate assessment of the negative and positive contributions to the RFC metrics can enhance the RFC sensitivity and clarify some of the mixed results in the literature regarding to the DMN and sensorimotor network involvement in adult aging.

The modulation of brain network integration and arousal during exploration

There is growing interest in how neuromodulators shape brain networks. Recent neuroimaging studies provide evidence that brainstem arousal systems, such as the locus coeruleus-norepinephrine system (LC-NE), influence functional connectivity and brain network topology, suggesting they have a role in flexibly reconfiguring brain networks in order to adapt behavior and cognition to environmental demands. To date, however, the relationship between brainstem arousal systems and functional connectivity has not been assessed within the context of a task with an established relationship between arousal and behavior, with most prior studies relying on incidental variations in arousal or pharmacological manipulation and static brain networks constructed over long periods of time. These factors have likely contributed to a heterogeneity of effects across studies. To address these issues, we took advantage of the association between LC-NE-linked arousal and exploration to probe the relationships between exploratory choice, arousal—as measured indirectly via pupil diameter—and brain network dynamics. Exploration in a bandit task was associated with a shift toward fewer, more weakly connected modules that were more segregated in terms of connectivity and topology but more integrated with respect to the diversity of cognitive systems represented in each module. Functional connectivity strength decreased, and changes in connectivity were correlated with changes in pupil diameter, in line with the hypothesis that brainstem arousal systems influence the dynamic reorganization of brain networks. More broadly, we argue that carefully aligning dynamic network analyses with task designs can increase the temporal resolution at which behaviorally- and cognitively-relevant modulations can be identified, and offer these results as a proof of concept of this approach.

Modulation of cortical slow oscillations and complexity across anesthesia levels

The ability of different groups of cortical neurons to engage in causal interactions that are at once differentiated and integrated results in complex dynamic patterns. Complexity is low during periods of unconsciousness (deep sleep, anesthesia, unresponsive wakefulness syndrome) in which the brain tends to generate a stereotypical pattern consisting of alternating active and silent periods of neural activity-slow oscillations- and is high during wakefulness. But how is cortical complexity built up? Is it a continuum? An open question is whether cortical complexity can vary within the same brain state. Here we recorded with 32-channel multielectrode arrays from the cortical surface of the mouse and used both spontaneous dynamics (wave propagation entropy and functional complexity) and a perturbational approach (a variation of the perturbation complexity index) to measure complexity at different anesthesia levels. Variations in anesthesia level within the bistable regime of slow oscillations (0.1-1.5 Hz) resulted in a modulation of the slow oscillation frequency. Both perturbational and spontaneous complexity increased with decreasing anesthesia levels, in correlation with the decrease in coherence of the underlying network. Changes in complexity level are related to, but not dependent on, changes in excitability. We conclude that cortical complexity can vary within a single brain state dominated by slow oscillations, building up to the higher complexity associated with consciousness.

Altered Neocortical Dynamics in a Mouse Model of Williams-Beuren Syndrome

Williams-Beuren syndrome (WBS) is a rare neurodevelopmental disorder characterized by moderate intellectual disability and learning difficulties alongside behavioral abnormalities such as hypersociability. Several structural and functional brain alterations are characteristic of this syndrome, as well as disturbed sleep and sleeping patterns. However, the detailed physiological mechanisms underlying WBS are mostly unknown. Here, we characterized the cortical dynamics in a mouse model of WBS previously reported to replicate most of the behavioral alterations described in humans. We recorded the laminar local field potential generated in the frontal cortex during deep anesthesia and characterized the properties of the emergent slow oscillation activity. Moreover, we performed micro-electrocorticogram recordings using multielectrode arrays covering the cortical surface of one hemisphere. We found significant differences between the cortical emergent activity and functional connectivity between wild-type mice and WBS model mice. Slow oscillations displayed Up states with diminished firing rate and lower high-frequency content in the gamma range. Lower firing rates were also recorded in the awake WBS animals while performing a marble burying task and could be associated with the decreased spine density and thus synaptic connectivity in this cortical area. We also found an overall increase in functional connectivity between brain areas, reflected in lower clustering and abnormally high integration, especially in the gamma range. These results expand previous findings in humans, suggesting that the cognitive deficits characterizing WBS might be associated with reduced excitability, plus an imbalance in the capacity to functionally integrate and segregate information.

Linking Entropy at Rest with the Underlying Structural Connectivity in the Healthy and Lesioned Brain

The brain is a network that mediates information processing through a wide range of states. The extent of state diversity is a reflection of the entropy of the network. Here we measured the entropy of brain regions (nodes) in empirical and modeled functional networks reconstructed from resting state fMRI to address the connection of entropy at rest with the underlying structure measured through diffusion spectrum imaging. Using 18 empirical and 18 modeled stroke networks, we also investigated the effect that focal lesions have on node entropy and information diffusion. Overall, positive correlations between node entropy and structure were observed, especially between node entropy and node strength in both empirical and modeled data. Although lesions were restricted to one hemisphere in all stroke patients, entropy reduction was not only present in nodes from the damaged hemisphere, but also in nodes from the contralesioned hemisphere, an effect replicated in modeled stroke networks. Globally, information diffusion was also affected in empirical and modeled strokes compared with healthy controls. This is the first study showing that artificial lesions affect local and global network aspects in very similar ways compared with empirical strokes, shedding new light into the functional nature of stroke.

Hierarchical Connectome Modes and Critical State Jointly Maximize Human Brain Functional Diversity

The brain requires diverse segregated and integrated processing to perform normal functions in terms of anatomical structure and self-organized dynamics with critical features, but the fundamental relationships between the complex structural connectome, critical state, and functional diversity remain unknown. Herein, we extend the eigenmode analysis to investigate the joint contribution of hierarchical modular structural organization and critical state to brain functional diversity. We show that the structural modes inherent to the hierarchical modular structural connectome allow a nested functional segregation and integration across multiple spatiotemporal scales. The real brain hierarchical modular organization provides large structural capacity for diverse functional interactions, which are generated by sequentially activating and recruiting the hierarchical connectome modes, and the critical state can best explore the capacity to maximize the functional diversity. Our results reveal structural and dynamical mechanisms that jointly support a balanced segregated and integrated brain processing with diverse functional interactions, and they also shed light on dysfunctional segregation and integration in neurodegenerative diseases and neuropsychiatric disorders.

Brain Map of Intrinsic Functional Flexibility in Anesthetized Monkeys and Awake Humans

Emerging neuroimaging studies emphasize the dynamic organization of spontaneous brain activity in both human and non-human primates, even under anesthesia. In a recent study, we were able to characterize the heterogeneous architecture of intrinsic functional flexibility in the awake, resting human brain using time-resolved analysis and a probabilistic model. However, it is unknown whether this organizational principle is preserved in the anesthetized monkey brain, and how anesthesia affects dynamic and static measurements of spontaneous brain activity. To investigate these issues, we collected resting-state functional magnetic resonance imaging (fMRI) datasets from 178 awake humans and 11 anesthetized monkeys (all healthy). Our recently established method, a complexity measurement (i.e., Shannon entropy) of dynamic functional connectivity patterns of each brain region, was used to map the intrinsic functional flexibility across the cerebral cortex. To further explore the potential effects of anesthesia, we performed time series analysis and correlation analysis between dynamic and static measurements within awake human and anesthetized monkey brains, respectively. We observed a heterogeneous profile of intrinsic functional flexibility in the anesthetized monkey brain, which showed some similarities to that of awake humans (r = 0.30, p = 0.007). However, we found that brain activity in anesthetized monkeys generally shifted toward random fluctuations. Moreover, there is a negative correlation between nodal entropy for the distribution of dynamic functional connectivity patterns and static functional connectivity strength in anesthetized monkeys, but not in awake humans. Our findings indicate that the heterogeneous architecture of intrinsic functional flexibility across cortex probably reflects an evolutionarily conserved aspect of functional brain organization, which persists across levels of cognitive processing (states of consciousness). The coupling between nodal entropy for the distribution of dynamic functional connectivity patterns and static functional connectivity strength may serve as a potential signature of anesthesia. This study not only offers fresh insight into the evolution of brain functional architecture, but also advances our understanding of the dynamics of spontaneous brain activity.

Mean-field analysis of directed modular networks

We considered a modular network with a binomial degree distribution and related the analytical relationships of the network properties (modularity, average clustering coefficient, and small-worldness) with structural parameters that define the network, i.e., number of nodes, number of modules, average node degree, and ratio of intra-modular to total connections. Even though modular networks are universally found in real-world systems and are consequently of broad interest in complex network science, the relationship between network properties and structural parameters has not yet been formulated. Here, we show that a series of equations for predicting the network properties can be related using a mean-field connectivity matrix that is defined on the basis of the structural parameters in the network generation algorithm. The theoretical results are then compared with values calculated numerically using the original connectivity matrix and are found to agree well, except when the connections between modules are sparse. Representation of the structure of the network using simple parameters is expected to be conducive for elucidating the structure-dynamics relationship.

Impact of modular organization on dynamical richness in cortical networks

As in many naturally formed networks, the brain exhibits an inherent modular architecture that is the basis of its rich operability, robustness, and integration-segregation capacity. However, the mechanisms that allow spatially segregated neuronal assemblies to swiftly change from localized to global activity remain unclear. Here, we integrate microfabrication technology with in vitro cortical networks to investigate the dynamical repertoire and functional traits of four interconnected neuronal modules. We show that the coupling among modules is central. The highest dynamical richness of the network emerges at a critical connectivity at the verge of physical disconnection. Stronger coupling leads to a persistently coherent activity among the modules, while weaker coupling precipitates the activity to be localized solely within the modules. An in silico modeling of the experiments reveals that the advent of coherence is mediated by a trade-off between connectivity and subquorum firing, a mechanism flexible enough to allow for the coexistence of both segregated and integrated activities. Our results unveil a new functional advantage of modular organization in complex networks of nonlinear units.

Criticality of the nonconservative earthquake model on random spatial networks

We study the nonconservative earthquake model on random spatial networks. The spatial networks are composed of sites on a two-dimensional (2D) plane which are connected locally. Differently from a regular lattice, the locations of sites are modeled in the way that sites are randomly placed on the plane. Using the same connectivity degree as a 2D lattice, however, the spatial network cannot exhibit critical earthquake behavior. Mimicking long range energy transfer, the connection radius is increased and the connectivity degree of the spatial network is increased. Then we show that the model exhibits self-organized criticality. The mechanism of the structural effect is presented. The spatial network includes many modules when connectivity degree is very small. The effect of modular structure on the avalanche dynamics is to limit the spreading of avalanches in the whole network. When the connectivity degree is larger, the long range energy transfer can overcome the effect of local modularity and criticality can be reached.

Effective Subnetwork Topology for Synchronizing Interconnected Networks of Coupled Phase Oscillators

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.

Functional complexity emerging from anatomical constraints in the brain: the significance of network modularity and rich-clubs

The large-scale structural ingredients of the brain and neural connectomes have been identified in recent years. These are, similar to the features found in many other real networks: the arrangement of brain regions into modules and the presence of highly connected regions (hubs) forming rich-clubs. Here, we examine how modules and hubs shape the collective dynamics on networks and we find that both ingredients lead to the emergence of complex dynamics. Comparing the connectomes of C. elegans, cats, macaques and humans to surrogate networks in which either modules or hubs are destroyed, we find that functional complexity always decreases in the perturbed networks. A comparison between simulated and empirically obtained resting-state functional connectivity indicates that the human brain, at rest, lies in a dynamical state that reflects the largest complexity its anatomical connectome can host. Last, we generalise the topology of neural connectomes into a new hierarchical network model that successfully combines modular organisation with rich-club forming hubs. This is achieved by centralising the cross-modular connections through a preferential attachment rule. Our network model hosts more complex dynamics than other hierarchical models widely used as benchmarks.

Functional brain networks: great expectations, hard times and the big leap forward

Many physical and biological systems can be studied using complex network theory, a new statistical physics understanding of graph theory. The recent application of complex network theory to the study of functional brain networks has generated great enthusiasm as it allows addressing hitherto non-standard issues in the field, such as efficiency of brain functioning or vulnerability to damage. However, in spite of its high degree of generality, the theory was originally designed to describe systems profoundly different from the brain. We discuss some important caveats in the wholesale application of existing tools and concepts to a field they were not originally designed to describe. At the same time, we argue that complex network theory has not yet been taken full advantage of, as many of its important aspects are yet to make their appearance in the neuroscience literature. Finally, we propose that, rather than simply borrowing from an existing theory, functional neural networks can inspire a fundamental reformulation of complex network theory, to account for its exquisitely complex functioning mode.

Rich club organization of macaque cerebral cortex and its role in network communication

Graph-theoretical analysis of brain connectivity data has revealed significant features of brain network organization across a range of species. Consistently, large-scale anatomical networks exhibit highly nonrandom attributes including an efficient small world modular architecture, with distinct network communities that are interlinked by hub regions. The functional importance of hubs motivates a closer examination of their mutual interconnections, specifically to examine the hypothesis that hub regions are more densely linked than expected based on their degree alone, i.e. forming a central rich club. Extending recent findings of rich club topology in the cat and human brain, this report presents evidence for the existence of rich club organization in the cerebral cortex of a non-human primate, the macaque monkey, based on a connectivity data set representing a collation of numerous tract tracing studies. Rich club regions comprise portions of prefrontal, parietal, temporal and insular cortex and are widely distributed across network communities. An analysis of network motifs reveals that rich club regions tend to form star-like configurations, indicative of their central embedding within sets of nodes. In addition, rich club nodes and edges participate in a large number of short paths across the network, and thus contribute disproportionately to global communication. As rich club regions tend to attract and disperse communication paths, many of the paths follow a characteristic pattern of first increasing and then decreasing node degree. Finally, the existence of non-reciprocal projections imposes a net directional flow of paths into and out of the rich club, with some regions preferentially attracting and others dispersing signals. Overall, the demonstration of rich club organization in a non-human primate contributes to our understanding of the network principles underlying neural connectivity in the mammalian brain, and further supports the hypothesis that rich club regions and connections have a central role in global brain communication.

Noise-induced synchronization in small world networks of phase oscillators

A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

Topological measure locating the effective crossover between segregation and integration in a modular network

We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hypergraph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intracluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed topological measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity and total integration, and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.

Characterizing the complexity of brain and mind networks

Recent studies of brain connectivity and language with methods of complex networks have revealed common features of organization. These observations open a window to better understand the intrinsic relationship between the brain and the mind by studying how information is either physically stored or mentally represented. In this paper, we review some of the results in both brain and linguistic networks, and we illustrate how modelling approaches can serve to comprehend the relationship between the structure of the brain and its function. On the one hand, we show that brain and neural networks display dynamical behaviour with optimal complexity in terms of a balance between their capacity to simultaneously segregate and integrate information. On the other hand, we show how principles of neural organization can be implemented into models of memory storage and recognition to reproduce spontaneous transitions between memories, resembling phenomena of memory association studied in psycholinguistic experiments.

Interplay between structure and dynamics in adaptive complex networks: emergence and amplification of modularity by adaptive dynamics

Many real networks display modular organization, which can influence dynamical clustering on the networks. Therefore, there have been proposals put forth recently to detect network communities by using dynamical clustering. In this paper, we study how the feedback from dynamical clusters can shape the network connection weights with a weight-adaptation scheme motivated from Hebbian learning in neural systems. We show that such a scheme generically leads to the formation of community structure in globally coupled chaotic oscillators. The number of communities in the adaptive network depends on coupling strength c and adaptation strength r. In a modular network, the adaptation scheme will enhance the intramodule connection weights and weaken the intermodule connection strengths, generating effectively separated dynamical clusters that coincide with the communities of the network. In this sense, the modularity of the network is amplified by the adaptation. Thus, for a network with a strong community structure, the adaptation scheme can evidently reflect its community structure by the resulting amplified weighted network. For a network with a weak community structure, the statistical properties of synchronization clusters from different realizations can be used to amplify the modularity of the communities so that they can be detected reliably by the other traditional algorithms.

Competition between intra-community and inter-community synchronization and relevance in brain cortical networks

In this paper the effects of inter-community links on the synchronization performance of community networks, especially on the competition between individual community and the whole network, are studied in detail. The study is organized from two aspects: the number or portion of inter-community links and the connection strategy of inter-community links between different communities. A critical point is found in the competition of global network and individual communities. Increasing the number of inter-community links will enhance the global synchronizability but degrade the synchronization performance of individual community before this point. After that the individual community will synchronize better and better as part of the whole network because the community structure is not so prominent. The critical point represents a balance region where the individual community is maximally independent while the information transmission remains effective between different communities. Among various connection strategies, connecting nodes belonging to different communities randomly rather than connecting nodes with larger degrees are the most efficient way to enhance global synchronization of the network. However, the dynamical modularity is the reverse case. A preferential connection scheme linking most of the hubs from the communities will allow rather efficient global synchronization while maintaining strong dynamical clustering of the communities. Interestingly, the observations are found to be relevant in a realistic network of cat cortex. The synchronization state is just at the critical point, which shows a reasonable combination of segregated function in individual communities and coordination among them. Our work sheds light on principles underlying the emergence of modular architectures in real network systems and provides guidance for the manipulation of synchronization in community networks.

Sustained activity in hierarchical modular neural networks: self-organized criticality and oscillations

Cerebral cortical brain networks possess a number of conspicuous features of structure and dynamics. First, these networks have an intricate, non-random organization. In particular, they are structured in a hierarchical modular fashion, from large-scale regions of the whole brain, via cortical areas and area subcompartments organized as structural and functional maps to cortical columns, and finally circuits made up of individual neurons. Second, the networks display self-organized sustained activity, which is persistent in the absence of external stimuli. At the systems level, such activity is characterized by complex rhythmical oscillations over a broadband background, while at the cellular level, neuronal discharges have been observed to display avalanches, indicating that cortical networks are at the state of self-organized criticality (SOC). We explored the relationship between hierarchical neural network organization and sustained dynamics using large-scale network modeling. Previously, it was shown that sparse random networks with balanced excitation and inhibition can sustain neural activity without external stimulation. We found that a hierarchical modular architecture can generate sustained activity better than random networks. Moreover, the system can simultaneously support rhythmical oscillations and SOC, which are not present in the respective random networks. The mechanism underlying the sustained activity is that each dense module cannot sustain activity on its own, but displays SOC in the presence of weak perturbations. Therefore, the hierarchical modular networks provide the coupling among subsystems with SOC. These results imply that the hierarchical modular architecture of cortical networks plays an important role in shaping the ongoing spontaneous activity of the brain, potentially allowing the system to take advantage of both the sensitivity of critical states and the predictability and timing of oscillations for efficient information processing.