Beware of the Small-World Neuroscientist!

The time-evolving epileptic brain network: concepts, definitions, accomplishments, perspectives

Epilepsy is now considered a network disease that affects the brain across multiple levels of spatial and temporal scales. The paradigm shift from an epileptic focus-a discrete cortical area from which seizures originate-to a widespread epileptic network-spanning lobes and hemispheres-considerably advanced our understanding of epilepsy and continues to influence both research and clinical treatment of this multi-faceted high-impact neurological disorder. The epileptic network, however, is not static but evolves in time which requires novel approaches for an in-depth characterization. In this review, we discuss conceptual basics of network theory and critically examine state-of-the-art recording techniques and analysis tools used to assess and characterize a time-evolving human epileptic brain network. We give an account on current shortcomings and highlight potential developments towards an improved clinical management of epilepsy.

Effect of network size on comparing different stock networks

We analyzed complex networks generated by the threshold method in the Korean and Indian stock markets during the non-crisis period of 2004 and the crisis period of 2008, while varying the size of the system. To create the stock network, we randomly selected N stock indices from the market and constructed the network based on cross-correlation among the time series of stock prices. We computed the average shortest path length L and average clustering coefficient C for several ensembles of generated stock networks and found that both metrics are influenced by network size. Since L and C are affected by network size N, a direct comparison of graph measures between stock networks with different numbers of nodes could lead to erroneous conclusions. However, we observed that the dependency of network measures on N is significantly reduced when comparing larger networks with normalized shortest path lengths. Additionally, we discovered that the effect of network size on network measures during the crisis period is almost negligible compared to the non-crisis periods.

Brain network communication: concepts, models and applications

Understanding communication and information processing in nervous systems is a central goal of neuroscience. Over the past two decades, advances in connectomics and network neuroscience have opened new avenues for investigating polysynaptic communication in complex brain networks. Recent work has brought into question the mainstay assumption that connectome signalling occurs exclusively via shortest paths, resulting in a sprawling constellation of alternative network communication models. This Review surveys the latest developments in models of brain network communication. We begin by drawing a conceptual link between the mathematics of graph theory and biological aspects of neural signalling such as transmission delays and metabolic cost. We organize key network communication models and measures into a taxonomy, aimed at helping researchers navigate the growing number of concepts and methods in the literature. The taxonomy highlights the pros, cons and interpretations of different conceptualizations of connectome signalling. We showcase the utility of network communication models as a flexible, interpretable and tractable framework to study brain function by reviewing prominent applications in basic, cognitive and clinical neurosciences. Finally, we provide recommendations to guide the future development, application and validation of network communication models.

A perturbation-based approach to identifying potentially superfluous network constituents

Constructing networks from empirical time-series data is often faced with the as yet unsolved issue of how to avoid potentially superfluous network constituents. Such constituents can result, e.g., from spatial and temporal oversampling of the system's dynamics, and neglecting them can lead to severe misinterpretations of network characteristics ranging from global to local scale. We derive a perturbation-based method to identify potentially superfluous network constituents that makes use of vertex and edge centrality concepts. We investigate the suitability of our approach through analyses of weighted small-world, scale-free, random, and complete networks.

Ordinal methods for a characterization of evolving functional brain networks

Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a resulting loss of information, this approach captures meaningful information about the temporal structure of the underlying system dynamics as well as about properties of interactions between coupled systems. This-together with its conceptual simplicity and robustness against measurement noise-makes ordinal time series analysis well suited to improve characterization of the still poorly understood spatiotemporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni- and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments, which would be necessary to advance characterization of evolving functional brain networks.

Network structure from a characterization of interactions in complex systems

Many natural and man-made complex dynamical systems can be represented by networks with vertices representing system units and edges the coupling between vertices. If edges of such a structural network are inaccessible, a widely used approach is to identify them with interactions between vertices, thereby setting up a functional network. However, it is an unsolved issue if and to what extent important properties of a functional network on the global and the local scale match those of the corresponding structural network. We address this issue by deriving functional networks from characterizing interactions in paradigmatic oscillator networks with widely-used time-series-analysis techniques for various factors that alter the collective network dynamics. Surprisingly, we find that particularly key constituents of functional networks—as identified with betweenness and eigenvector centrality—coincide with ground truth to a high degree, while global topological and spectral properties—clustering coefficient, average shortest path length, assortativity, and synchronizability—clearly deviate. We obtain similar concurrences for an empirical network. Our findings are of relevance for various scientific fields and call for conceptual and methodological refinements to further our understanding of the relationship between structure and function of complex dynamical systems.

From Coherence to Multivariate Causal Estimators of EEG Connectivity

The paper concerns the development of methods of EEG functional connectivity estimation including short overview of the currently applied measures describing their advantages and flaws. Linear and non-linear, bivariate and multivariate methods are confronted. The performance of different connectivity measures in respect of robustness to noise, common drive effect and volume conduction is considered providing a guidance towards future developments in the field, which involve evaluation not only functional, but also effective (causal) connectivity. The time-varying connectivity measure making possible estimation of dynamical information processing in brain is presented. The methods of post-processing of connectivity results are considered involving application of advanced graph analysis taking into account community structure of networks and providing hierarchy of networks rather than the single, binary networks currently used.

Network dynamics scale with levels of awareness

Small world topologies are thought to provide a valuable insight into human brain organisation and consciousness. However, functional magnetic resonance imaging studies in consciousness have not yielded consistent results. Given the importance of dynamics for both consciousness and cognition, here we investigate how the diversity of small world dynamics (quantified by sample entropy; dSW-E¹) scales with decreasing levels of awareness (i.e., sedation and disorders of consciousness). Paying particular attention to result reproducibility, we show that dSW-E is a consistent predictor of levels of awareness even when controlling for the underlying functional connectivity dynamics. We find that dSW-E of subcortical and cortical areas are predictive, with the former showing higher and more robust effect sizes across analyses. We find that the network dynamics of intermodular communication in the cerebellum also have unique predictive power for levels of awareness. Consequently, we propose that the dynamic reorganisation of the functional information architecture, in particular of the subcortex, is a characteristic that emerges with awareness and has explanatory power beyond that of the complexity of dynamic functional connectivity.

Mental Calculation Drives Reliable and Weak Distant Connectivity While Music Listening Induces Dense Local Connectivity

Mathematical calculation usually requires sustained attention to manipulate numbers in the mind, while listening to light music has a relaxing effect on the brain. The differences in the corresponding brain functional network topologies underlying these behaviors remain rarely known. Here, we systematically examined the brain dynamics of four behaviors (resting with eyes closed and eyes open, tasks of music listening and mental calculation) using 64-channel electroencephalogram (EEG) recordings and graph theory analysis. We developed static and dynamic minimum spanning tree (MST) analysis method and demonstrated that the brain network topology under mental calculation is a more line-like structure with less tree hierarchy and leaf fraction; however, the hub regions, which are mainly located in the frontal, temporal and parietal regions, grow more stable over time. In contrast, music-listening drives the brain to exhibit a highly rich network of star structure, and the hub regions are mainly located in the posterior regions. We then adopted the dynamic dissimilarity of different MSTs over time based on the graph Laplacian and revealed low dissimilarity during mental calculation. These results suggest that the human brain functional connectivity of individuals has unique dynamic diversity and flexibility under various behaviors.

Supplementary Information

The online version contains supplementary material available at 10.1007/s43657-021-00027-w.

The organization of functional neurocognitive networks in focal epilepsy correlates with domain-specific cognitive performance

Understanding and diagnosing cognitive impairment in epilepsy remains a prominent challenge. New etiological models suggest that cognitive difficulties might not be directly linked to seizure activity, but are rather a manifestation of a broader brain pathology. Consequently, treating seizures is not sufficient to alleviate cognitive symptoms, highlighting the need for novel diagnostic tools. Here, we investigated whether the organization of three intrinsic, resting-state functional connectivity networks was correlated with domain-specific cognitive test performance. Using individualized EEG source reconstruction and graph theory, we examined the association between network small worldness and cognitive test performance in 23 patients with focal epilepsy and 17 healthy controls, who underwent a series of standardized pencil-and-paper and digital cognitive tests. We observed that the specific networks robustly correlated with test performance in distinct cognitive domains. Specifically, correlations were evident between the default mode network and memory in patients, the central-executive network and executive functioning in controls, and the salience network and social cognition in both groups. Interestingly, the correlations were evident in both groups, but in different domains, suggesting an alteration in these functional neurocognitive networks in focal epilepsy. The present findings highlight the potential clinical relevance of functional brain network dysfunction in cognitive impairment.

Brain network integration dynamics are associated with loss and recovery of consciousness induced by sevoflurane

The dynamic interplay of integration and segregation in the brain is at the core of leading theoretical accounts of consciousness. The human brain dynamically alternates between a sub-state where integration predominates, and a predominantly segregated sub-state, with different roles in supporting cognition and behaviour. Here, we combine graph theory and dynamic functional connectivity to compare resting-state functional MRI data from healthy volunteers before, during, and after loss of responsiveness induced with different concentrations of the inhalational anaesthetic, sevoflurane. We show that dynamic states characterised by high brain integration are especially vulnerable to general anaesthesia, exhibiting attenuated complexity and diminished small-world character. Crucially, these effects are reversed upon recovery, demonstrating their association with consciousness. Higher doses of sevoflurane (3% vol and burst-suppression) also compromise the temporal balance of integration and segregation in the human brain. Additionally, we demonstrate that reduced anticorrelations between the brain's default mode and executive control networks dynamically reconfigure depending on the brain's state of integration or segregation. Taken together, our results demonstrate that the integrated sub-state of brain connectivity is especially vulnerable to anaesthesia, in terms of both its complexity and information capacity, whose breakdown represents a generalisable biomarker of loss of consciousness and its recovery.

Source-level EEG and graph theory reveal widespread functional network alterations in focal epilepsy

OBJECTIVE

The hypersynchronous neuronal activity associated with epilepsy causes widespread functional network disruptions extending beyond the epileptogenic zone. This altered network topology is considered a mediator for non-seizure symptoms, such as cognitive impairment. The aim of this study was to investigate functional network alterations in focal epilepsy patients with good seizure control and high quality of life.

METHODS

We compared twenty-two focal epilepsy patients and sixteen healthy controls on graph metrics derived from functional connectivity of source-level resting-state EEG. Graph metrics were calculated over a range of network densities in five frequency bands.

RESULTS

We observed a significantly increased small world index in patients relative to controls. On the local level, two left-hemisphere regions displayed a shift towards greater alpha band "hubness". The findings were not mediated by age, sex or education, nor by age of epilepsy onset, duration or focus lateralisation.

CONCLUSIONS

Widespread functional network alterations are evident in focal epilepsy, even in a cohort characterised by successful anti-seizure medication therapy and high quality of life. These findings might support the position that functional network analysis could hold clinical relevance for epilepsy.

SIGNIFICANCE

Focal epilepsy is accompanied by global and local functional network aberrancies which might be implied in the sustenance of non-seizure symptoms.

Inferring network properties from time series using transfer entropy and mutual information: Validation of multivariate versus bivariate approaches

Functional and effective networks inferred from time series are at the core of network neuroscience. Interpreting properties of these networks requires inferred network models to reflect key underlying structural features. However, even a few spurious links can severely distort network measures, posing a challenge for functional connectomes. We study the extent to which micro- and macroscopic properties of underlying networks can be inferred by algorithms based on mutual information and bivariate/multivariate transfer entropy. The validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Simulations are based on a neural mass model and on autoregressive dynamics (employing Gaussian estimators for direct comparison to functional connectivity and Granger causality). We find that multivariate transfer entropy captures key properties of all network structures for longer time series. Bivariate methods can achieve higher recall (sensitivity) for shorter time series but are unable to control false positives (lower specificity) as available data increases. This leads to overestimated clustering, small-world, and rich-club coefficients, underestimated shortest path lengths and hub centrality, and fattened degree distribution tails. Caution should therefore be used when interpreting network properties of functional connectomes obtained via correlation or pairwise statistical dependence measures, rather than more holistic (yet data-hungry) multivariate models.

We compare bivariate and multivariate methods for inferring networks from time series, which are generated using a neural mass model and autoregressive dynamics. We assess their ability to reproduce key properties of the underlying structural network. Validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Even a few spurious links can severely bias key network properties. Multivariate transfer entropy performs best on all topologies for longer time series.

Topological Structure and Robustness of the Lymph Node Conduit System

Fibroblastic reticular cells (FRCs) form a road-like cellular network in lymph nodes (LNs) that provides essential chemotactic, survival, and regulatory signals for immune cells. While the topological characteristics of the FRC network have been elaborated, the network properties of the micro-tubular conduit system generated by FRCs, which drains lymph fluid through a pipeline-like system to distribute small molecules and antigens, has remained unexplored. Here, we quantify the crucial 3D morphometric parameters and determine the topological properties governing the structural organization of the intertwined networks. We find that the conduit system exhibits lesser small-worldness and lower resilience to perturbation compared to the FRC network, while the robust topological organization of both networks is maintained in a lymphotoxin-β-receptor-independent manner. Overall, the high-resolution topological analysis of the "roads-and-pipes" networks highlights essential parameters underlying the functional organization of LN micro-environments and will, hence, advance the development of multi-scale LN models.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena

We study patterns of partial synchronization in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay between dynamics and network topology, we perform long-term simulations of oscillatory dynamics on different paradigmatic network structures: random networks, regular nonlocally coupled ring networks, ring networks with fractal connectivities, and small-world networks with various rewiring probability. Among these networks, a small-world network with intermediate rewiring probability best mimics the findings achieved with the simulations using the empirical structural connectivity. For the other network topologies, either no spontaneously occurring epileptic-seizure-related synchronization phenomena can be observed in the simulated dynamics, or the overall degree of synchronization remains high throughout the simulation. This indicates that a topology with some balance between regularity and randomness favors the self-initiation and self-termination of episodes of seizure-like strong synchronization.

Von Interaktionen zu Interaktionsnetzwerken: Zeitabhängige funktionelle Netzwerke am Beispiel der Epilepsie

Das menschliche Gehirn ist ein komplexes Netzwerk aus interagierenden nichtstationären Subsystemen (Netzwerk von Netzwerken), deren komplizierte räumlich-zeitliche Dynamiken bis heute nur unzureichend verstanden sind. Dabei versprechen aktuelle Entwicklungen im Bereich der Zeitreihenanalyse sowie der Theorie komplexer Netzwerke neue und verbesserte Einblicke in die Dynamiken von Hirnnetzwerken auf verschiedenen räumlich-zeitlichen Skalen. Wir geben einen Überblick über diese Entwicklungen und besprechen am Beispiel zeitabhängiger epileptischer Hirnnetzwerke Fortschritte im Verständnis von Hirndynamiken, die über multiple Skalen hinweg variieren.

Functional Connectome Analyses Reveal the Human Olfactory Network Organization

The olfactory system is uniquely heterogeneous, performing multifaceted functions (beyond basic sensory processing) across diverse, widely distributed neural substrates. While knowledge of human olfaction continues to grow, it remains unclear how the olfactory network is organized to serve this unique set of functions.

The olfactory system is uniquely heterogeneous, performing multifaceted functions (beyond basic sensory processing) across diverse, widely distributed neural substrates. While knowledge of human olfaction continues to grow, it remains unclear how the olfactory network is organized to serve this unique set of functions. Leveraging a large and high-quality resting-state functional magnetic resonance imaging (rs-fMRI) dataset of nearly 900 participants from the Human Connectome Project (HCP), we identified a human olfactory network encompassing cortical and subcortical regions across the temporal and frontal lobes. Highlighting its reliability and generalizability, the connectivity matrix of this olfactory network mapped closely onto that extracted from an independent rs-fMRI dataset. Graph theoretical analysis further explicated the organizational principles of the network. The olfactory network exhibits a modular composition of three (i.e., the sensory, limbic, and frontal) subnetworks and demonstrates strong small-world properties, high in both global integration and local segregation (i.e., circuit specialization). This network organization thus ensures the segregation of local circuits, which are nonetheless integrated via connecting hubs [i.e., amygdala (AMY) and anterior insula (INSa)], thereby enabling the specialized, yet integrative, functions of olfaction. In particular, the degree of local segregation positively predicted olfactory discrimination performance in the independent sample, which we infer as a functional advantage of the network organization. In sum, an olfactory functional network has been identified through the large HCP dataset, affording a representative template of the human olfactory functional neuroanatomy. Importantly, the topological analysis of the olfactory network provides network-level insights into the remarkable functional specialization and spatial segregation of the olfactory system.

Rich-club in the brain's macrostructure: Insights from graph theoretical analysis

The brain is a complex network. Growing evidence supports the critical roles of a set of brain regions within the brain network, known as the brain’s cores or hubs. These regions require high energy cost but possess highly efficient neural information transfer in the brain’s network and are termed the rich-club. The rich-club of the brain network is essential as it directly regulates functional integration across multiple segregated regions and helps to optimize cognitive processes. Here, we review the recent advances in rich-club organization to address the fundamental roles of the rich-club in the brain and discuss how these core brain regions affect brain development and disorders. We describe the concepts of the rich-club behind network construction in the brain using graph theoretical analysis. We also highlight novel insights based on animal studies related to the rich-club and illustrate how human studies using neuroimaging techniques for brain development and psychiatric/neurological disorders may be relevant to the rich-club phenomenon in the brain network.

Improved functional connectivity network estimation for brain networks using multivariate partial coherence

OBJECTIVE

Graphical networks and network metrics are widely used to understand and characterise brain networks and brain function. These methods can be applied to a range of electrophysiological data including electroencephalography, local field potential and single unit recordings. Functional networks are often constructed using pair-wise correlation between variables. The objective of this study is to demonstrate that functional networks can be more accurately estimated using partial correlation than with pair-wise correlation.

APPROACH

We compared network metrics derived from unconditional and conditional graphical networks, obtained using coherence and multivariate partial coherence (MVPC), respectively. Graphical networks were constructed using coherence and MVPC estimates, and binary and weighted network metrics derived from these: node degree, path length, clustering coefficients and small-world index.

MAIN RESULTS

Network metrics were applied to simulated and experimental single unit spike train data. Simulated data used a 10x10 grid of simulated cortical neurons with centre-surround connectivity. Conditional network metrics gave a more accurate representation of the known connectivity: Numbers of excitatory connections had range 3-11, unconditional binary node degree had range 6-80, conditional node degree had range 2-13. Experimental data used multi-electrode array recording with 19 single-units from left and right hippocampal brain areas in a rat model for epilepsy. Conditional network analysis showed similar trends to simulated data, with lower binary node degree and longer binary path lengths compared to unconditional networks.

SIGNIFICANCE

We conclude that conditional networks, where common dependencies are removed through partial coherence analysis, give a more accurate representation of the interactions in a graphical network model. These results have important implications for graphical network analyses of brain networks and suggest that functional networks should be derived using partial correlation, based on MVPC estimates, as opposed to the common approach of pair-wise correlation.

Multiscale Community Detection in Functional Brain Networks Constructed Using Dynamic Time Warping

Previous studies have focused on the detection of community structures of brain networks constructed with resting-state functional magnetic resonance imaging (fMRI) data. Pearson correlation is often used to describe the connections between nodes in the construction of functional brain networks, which typically ignores the inherent timing and validity of fMRI time series. To solve this problem, this study applied the Dynamic Time Warp (DTW) algorithm to determine the correlation between two brain regions by comparing the synchronization and asynchrony of the time series. In addition, to determine the best community structure for each subject, we further divided the brain network into different scales, and then detected the different communities in these brain networks by using Modularity, Variation of Information (VI) and Normalized Mutual Information (NMI) as structural monitoring variables. Finally, we affirmed each subject's best community structure based on them. The experiments showed that through the method proposed in this paper, we not only accurately discovered important components of seven basic functional subnetworks, but also found that the putamen and Heschl's gyrus have a relationship with the inferior parietal network. Most importantly, this method can also determine each subject's functional brain network density, thus confirming the findings of studies testing real brain networks.

Consciousness-specific dynamic interactions of brain integration and functional diversity

Prominent theories of consciousness emphasise different aspects of neurobiology, such as the integration and diversity of information processing within the brain. Here, we combine graph theory and dynamic functional connectivity to compare resting-state functional MRI data from awake volunteers, propofol-anaesthetised volunteers, and patients with disorders of consciousness, in order to identify consciousness-specific patterns of brain function. We demonstrate that cortical networks are especially affected by loss of consciousness during temporal states of high integration, exhibiting reduced functional diversity and compromised informational capacity, whereas thalamo-cortical functional disconnections emerge during states of higher segregation. Spatially, posterior regions of the brain’s default mode network exhibit reductions in both functional diversity and integration with the rest of the brain during unconsciousness. These results show that human consciousness relies on spatio-temporal interactions between brain integration and functional diversity, whose breakdown may represent a generalisable biomarker of loss of consciousness, with potential relevance for clinical practice.

How do diversity (entropy) and integration of activity across brain regions interact to support consciousness? Here the authors show that anaesthetised individuals and patients with disorders of consciousness exhibit overlapping reductions in both diversity and integration in the brain’s default mode network.

Increased segregation of functional networks in developing brains

A growing literature conceptualises typical brain development from a network perspective. However, largely due to technical and methodological challenges inherent in paediatric functional neuroimaging, there remains an important gap in our knowledge regarding the typical development of functional brain networks in "preschool" childhood (i.e., children younger than 6 years of age). In this study, we recorded brain oscillatory activity using age-appropriate magnetoencephalography in 24 children, including 14 preschool children aged from 4 to 6 years and 10 school children aged from 7 to 12 years. We compared the topology of the resting-state brain networks in these children, estimated using minimum spanning tree (MST) constructed from phase synchrony between beamformer-reconstructed time-series, with that of 24 adults. Our results show that during childhood the MST topology shifts from a star-like (centralised) toward a more line-like (de-centralised) configuration, indicating the functional brain networks become increasingly segregated. In addition, the increasing global network segregation is frequency-independent and accompanied by decreases in centrality (or connectedness) of cortical regions with age, especially in areas of the default mode network. We propose a heuristic MST model of "network space", which posits a clear developmental trajectory for the emergence of complex brain networks. Our results not only revealed topological reorganisation of functional networks across multiple temporal and spatial scales in childhood, but also fill a gap in the literature regarding neurophysiological mechanisms of functional brain maturation during the preschool years of childhood.

Brain synchronizability, a false friend

Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network is, given its topological organization, are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. Here, we argue that brain dynamics does not fulfil the formal criteria under which synchronizability is usually quantified and, perhaps more importantly, this measure refers to a global dynamical condition that never holds in the brain (not even in the most pathological conditions), and therefore no neurophysiological conclusions should be drawn based on it. We discuss the meaning of synchronizability and its applicability to neuroscience and propose alternative ways to quantify brain networks synchronization.

The structural connectome in traumatic brain injury: A meta-analysis of graph metrics

Although recent structural connectivity studies of traumatic brain injury (TBI) have used graph theory to evaluate alterations in global integration and functional segregation, pooled analysis is needed to examine the robust patterns of change in graph metrics across studies. Following a systematic search, 15 studies met the inclusion criteria for review. Of these, ten studies were included in a random-effects meta-analysis of global graph metrics, and subgroup analyses examined the confounding effects of severity and time since injury. The meta-analysis revealed significantly higher values of normalised clustering coefficient (gö=ö1.445, CI=[0.512, 2.378], pö=ö0.002) and longer characteristic path length (gö=ö0.514, CI=[0.190, 0.838], pö=ö0.002) in TBI patients compared with healthy controls. Our findings suggest that the TBI structural network has shifted away from the balanced small-world network towards a regular lattice. Therefore, these graph metrics may be useful markers of neurocognitive dysfunction in TBI. We conclude that the pattern of change revealed by our analysis should be used to guide hypothesis-driven research into the role of graph metrics as diagnostic and prognostic biomarkers.

Topological structures are consistently overestimated in functional complex networks

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

Stability of graph theoretical measures in structural brain networks in Alzheimer's disease

Graph analysis has become a popular approach to study structural brain networks in neurodegenerative disorders such as Alzheimer's disease (AD). However, reported results across similar studies are often not consistent. In this paper we investigated the stability of the graph analysis measures clustering, path length, global efficiency and transitivity in a cohort of AD (N = 293) and control subjects (N = 293). More specifically, we studied the effect that group size and composition, choice of neuroanatomical atlas, and choice of cortical measure (thickness or volume) have on binary and weighted network properties and relate them to the magnitude of the differences between groups of AD and control subjects. Our results showed that specific group composition heavily influenced the network properties, particularly for groups with less than 150 subjects. Weighted measures generally required fewer subjects to stabilize and all assessed measures showed robust significant differences, consistent across atlases and cortical measures. However, all these measures were driven by the average correlation strength, which implies a limitation of capturing more complex features in weighted networks. In binary graphs, significant differences were only found in the global efficiency and transitivity measures when using cortical thickness measures to define edges. The findings were consistent across the two atlases, but no differences were found when using cortical volumes. Our findings merits future investigations of weighted brain networks and suggest that cortical thickness measures should be preferred in future AD studies if using binary networks. Further, studying cortical networks in small cohorts should be complemented by analyzing smaller, subsampled groups to reduce the risk that findings are spurious.

Nonlinearity in stock networks

Stock networks, constructed from stock price time series, are a well-established tool for the characterization of complex behavior in stock markets. Following Mantegna's seminal paper, the linear Pearson's correlation coefficient between pairs of stocks has been the usual way to determine network edges. Recently, possible effects of nonlinearity on the graph-theoretical properties of such networks have been demonstrated when using nonlinear measures such as mutual information instead of linear correlation. In this paper, we quantitatively characterize the nonlinearity in stock time series and the effect it has on stock network properties. This is achieved by a systematic multi-step approach that allows us to quantify the nonlinearity of coupling; correct its effects wherever it is caused by simple univariate non-Gaussianity; potentially localize in space and time any remaining strong sources of this nonlinearity; and, finally, study the effect nonlinearity has on global network properties. By applying this multi-step approach to stocks included in three prominent indices (New York Stock Exchange 100, Financial Times Stock Exchange 100, and Standard & Poor 500), we establish that the apparent nonlinearity that has been observed is largely due to univariate non-Gaussianity. Furthermore, strong nonstationarity in a few specific stocks may play a role. In particular, the sharp decrease in some stocks during the global financial crisis of 2008 gives rise to apparent nonlinear dependencies among stocks.

Impact of type of intracranial EEG sensors on link strengths of evolving functional brain networks

Objective and Approach: Investigating properties of evolving functional brain networks has become a valuable tool to characterize the complex dynamics of the epileptic brain. Such networks are usually derived from electroencephalograms (EEG) recorded with sensors implanted chronically into deeper structures of the brain and/or placed onto the cortex. It is still unclear, however, whether the use of different sensors for an identification of network nodes affects properties of functional brain networks. We address this question by investigating properties of links of such networks that we characterize by assessing interactions in multi-sensor, multi-day EEG data recorded from 49 epilepsy patients during presurgical evaluation. These data allow us to study the impact of different types of sensors together with the impact of various physiologic and pathophysiologic activities on the properties of links.

MAIN RESULTS

We observe that different types of sensors differently impact on spatial means and temporal fluctuations of link strengths. Moreover, the impact depends on the relative anatomical location of sensors with respect to location and extent of sources of the prevailing activities.

SIGNIFICANCE

Type and location of sensors should be considered when constructing networks.

Computational Neuroscience: Mathematical and Statistical Perspectives

Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm.

On the Reliability of Individual Brain Activity Networks

There is intense interest in fMRI research on whole-brain functional connectivity, and however, two fundamental issues are still unresolved: the impact of spatiotemporal data resolution (spatial parcellation and temporal sampling) and the impact of the network construction method on the reliability of functional brain networks. In particular, the impact of spatiotemporal data resolution on the resulting connectivity findings has not been sufficiently investigated. In fact, a number of studies have already observed that functional networks often give different conclusions across different parcellation scales. If the interpretations from functional networks are inconsistent across spatiotemporal scales, then the whole validity of the functional network paradigm is called into question. This paper investigates the consistency of resting state network structure when using different temporal sampling or spatial parcellation, or different methods for constructing the networks. To pursue this, we develop a novel network comparison framework based on persistent homology from a topological data analysis. We use the new network comparison tools to characterize the spatial and temporal scales under which consistent functional networks can be constructed. The methods are illustrated on Human Connectome Project data, showing that the DISCOH₂ network construction method outperforms other approaches at most data spatiotemporal resolutions.

Surrogate-assisted identification of influences of network construction on evolving weighted functional networks

We aim at identifying factors that may affect the characteristics of evolving weighted networks derived from empirical observations. To this end, we employ various chains of analysis that are often used in field studies for a data-driven derivation and characterization of such networks. As an example, we consider fully connected, weighted functional brain networks before, during, and after epileptic seizures that we derive from multichannel electroencephalographic data recorded from epilepsy patients. For these evolving networks, we estimate clustering coefficient and average shortest path length in a time-resolved manner. Lastly, we make use of surrogate concepts that we apply at various levels of the chain of analysis to assess to what extent network characteristics are dominated by properties of the electroencephalographic recordings and/or the evolving weighted networks, which may be accessible more easily. We observe that characteristics are differently affected by the unavoidable referencing of the electroencephalographic recording, by the time-series-analysis technique used to derive the properties of network links, and whether or not networks were normalized. Importantly, for the majority of analysis settings, we observe temporal evolutions of network characteristics to merely reflect the temporal evolutions of mean interaction strengths. Such a property of the data may be accessible more easily, which would render the weighted network approach-as used here-as an overly complicated description of simple aspects of the data.

Biological conservation law as an emerging functionality in dynamical neuronal networks

Scientists strive to understand how functionalities, such as conservation laws, emerge in complex systems. Living complex systems in particular create high-ordered functionalities by pairing up low-ordered complementary processes, e.g., one process to build and the other to correct. We propose a network mechanism that demonstrates how collective statistical laws can emerge at a macro (i.e., whole-network) level even when they do not exist at a unit (i.e., network-node) level. Drawing inspiration from neuroscience, we model a highly stylized dynamical neuronal network in which neurons fire either randomly or in response to the firing of neighboring neurons. A synapse connecting two neighboring neurons strengthens when both of these neurons are excited and weakens otherwise. We demonstrate that during this interplay between the synaptic and neuronal dynamics, when the network is near a critical point, both recurrent spontaneous and stimulated phase transitions enable the phase-dependent processes to replace each other and spontaneously generate a statistical conservation law-the conservation of synaptic strength. This conservation law is an emerging functionality selected by evolution and is thus a form of biological self-organized criticality in which the key dynamical modes are collective.

Small-world bias of correlation networks: From brain to climate

Complex systems are commonly characterized by the properties of their graph representation. Dynamical complex systems are then typically represented by a graph of temporal dependencies between time series of state variables of their subunits. It has been shown recently that graphs constructed in this way tend to have relatively clustered structure, potentially leading to spurious detection of small-world properties even in the case of systems with no or randomly distributed true interactions. However, the strength of this bias depends heavily on a range of parameters and its relevance for real-world data has not yet been established. In this work, we assess the relevance of the bias using two examples of multivariate time series recorded in natural complex systems. The first is the time series of local brain activity as measured by functional magnetic resonance imaging in resting healthy human subjects, and the second is the time series of average monthly surface air temperature coming from a large reanalysis of climatological data over the period 1948-2012. In both cases, the clustering in the thresholded correlation graph is substantially higher compared with a realization of a density-matched random graph, while the shortest paths are relatively short, showing thus distinguishing features of small-world structure. However, comparable or even stronger small-world properties were reproduced in correlation graphs of model processes with randomly scrambled interconnections. This suggests that the small-world properties of the correlation matrices of these real-world systems indeed do not reflect genuinely the properties of the underlying interaction structure, but rather result from the inherent properties of correlation matrix.

Hyperglycemia Reduces Efficiency of Brain Networks in Subjects with Type 2 Diabetes

Previous research has shown that the brain is an important target of diabetic complications. Since brain regions are interconnected to form a large-scale neural network, we investigated whether severe hyperglycemia affects the topology of the brain network in people with type 2 diabetes. Twenty middle-aged (average age: 54 years) individuals with poorly controlled diabetes (HbA1c: 8.9−14.6%, 74−136 mmol/mol) and 20 age-, sex-, and education-matched healthy volunteers were recruited. Graph theoretic network analysis was performed with axonal fiber tractography and tract-based spatial statistics (TBSS) using diffusion tensor imaging. Associations between the blood glucose level and white matter network characteristics were investigated. Individuals with diabetes had lower white matter network efficiency (P<0.001) and longer white matter path length (P<0.05) compared to healthy individuals. Higher HbA1c was associated with lower network efficiency (r = −0.53, P = 0.001) and longer network path length (r = 0.40, P<0.05). A disruption in local microstructural integrity was found in the multiple white matter regions and associated with higher HbA1c and fasting plasma glucose levels (corrected P<0.05). Poorer glycemic control is associated with lower efficiency and longer connection paths of the global brain network in individuals with diabetes. Chronic hyperglycemia in people with diabetes may disrupt the brain’s topological integration, and lead to mental slowing and cognitive impairment.