Dynamical reconnection and stability constraints on cortical network architecture

Spontaneous Activity Predicts Survival of Developing Cortical Neurons

Spontaneous activity plays a crucial role in brain development by coordinating the integration of immature neurons into emerging cortical networks. High levels and complex patterns of spontaneous activity are generally associated with low rates of apoptosis in the cortex. However, whether spontaneous activity patterns directly encode for survival of individual cortical neurons during development remains an open question. Here, we longitudinally investigated spontaneous activity and apoptosis in developing cortical cultures, combining extracellular electrophysiology with calcium imaging. These experiments demonstrated that the early occurrence of calcium transients was strongly linked to neuronal survival. Silent neurons exhibited a higher probability of cell death, whereas high frequency spiking and burst behavior were almost exclusively detected in surviving neurons. In local neuronal clusters, activity of neighboring neurons exerted a pro-survival effect, whereas on the functional level, networks with a high modular topology were associated with lower cell death rates. Using machine learning algorithms, cell fate of individual neurons was predictable through the integration of spontaneous activity features. Our results indicate that high frequency spiking activity constrains apoptosis in single neurons through sustained calcium rises and thereby consolidates networks in which a high modular topology is reached during early development.

Long-range temporal correlations in scale-free neuromorphic networks

Biological neuronal networks are the computing engines of the mammalian brain. These networks exhibit structural characteristics such as hierarchical architectures, small-world attributes, and scale-free topologies, providing the basis for the emergence of rich temporal characteristics such as scale-free dynamics and long-range temporal correlations. Devices that have both the topological and the temporal features of a neuronal network would be a significant step toward constructing a neuromorphic system that can emulate the computational ability and energy efficiency of the human brain. Here we use numerical simulations to show that percolating networks of nanoparticles exhibit structural properties that are reminiscent of biological neuronal networks, and then show experimentally that stimulation of percolating networks by an external voltage stimulus produces temporal dynamics that are self-similar, follow power-law scaling, and exhibit long-range temporal correlations. These results are expected to have important implications for the development of neuromorphic devices, especially for those based on the concept of reservoir computing.

Biological neuronal networks exhibit well-defined properties such as hierarchical structures and scale-free topologies, as well as a high degree of local clustering and short path lengths between nodes. These structural properties are intimately connected to the observed long-range temporal correlations in the network dynamics. Fabrication of artificial networks with similar structural properties would facilitate brain-like (“neuromorphic”) computing. Here we show experimentally that percolating networks of nanoparticles exhibit similar long-range temporal correlations to those of biological neuronal networks and use simulations to demonstrate that the dynamics arise from an underlying scale-free network architecture. We discuss similarities between the biological and percolating systems and highlight the potential for the percolating networks to be used in neuromorphic computing applications.

The road ahead in clinical network neuroscience

Clinical network neuroscience, the study of brain network topology in neurological and psychiatric diseases, has become a mainstay field within clinical neuroscience. Being a multidisciplinary group of clinical network neuroscience experts based in The Netherlands, we often discuss the current state of the art and possible avenues for future investigations. These discussions revolve around questions like "How do dynamic processes alter the underlying structural network?" and "Can we use network neuroscience for disease classification?" This opinion paper is an incomplete overview of these discussions and expands on ten questions that may potentially advance the field. By no means intended as a review of the current state of the field, it is instead meant as a conversation starter and source of inspiration to others.

Determination of effective brain connectivity from activity correlations

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

Memory systems 2018 - Towards a new paradigm

The multiple memory systems theory (MMS) postulates that the brain stores information based on the independent and parallel activity of a number of modules, each with distinct properties, dynamics, and neural basis. Much of the evidence for this theory comes from dissociation studies indicating that damage to restricted brain areas cause selective types of memory deficits. MMS has been the prevalent paradigm in memory research for more than thirty years, even as it has been adjusted several times to accommodate new data. However, recent empirical results indicating that the memory systems are not always dissociable constitute a challenge to fundamental tenets of the current theory because they suggest that representations formed by individual memory systems can contribute to more than one type of memory-driven behavioral strategy. This problem can be addressed by applying a dynamic network perspective to memory architecture. According to this view, memory networks can reconfigure or transiently couple in response to environmental demands. Within this context, the neural network underlying a specific memory system can act as an independent unit or as an integrated component of a higher order meta-network. This dynamic network model proposes a way in which empirical evidence that challenges the idea of distinct memory systems can be incorporated within a modular memory architecture. The model also provides a framework to account for the complex interactions among memory systems demonstrated at the behavioral level. Advances in the study of dynamic networks can generate new ideas to experimentally manipulate and control memory in basic or clinical research.

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.

A mechanistic model of connector hubs, modularity and cognition

The human brain network is modular-comprised of communities of tightly interconnected nodes1. This network contains local hubs, which have many connections within their own communities, and connector hubs, which have connections diversely distributed across communities2,3. A mechanistic understanding of these hubs and how they support cognition has not been demonstrated. Here, we leveraged individual differences in hub connectivity and cognition. We show that a model of hub connectivity accurately predicts the cognitive performance of 476 individuals in four distinct tasks. Moreover, there is a general optimal network structure for cognitive performance-individuals with diversely connected hubs and consequent modular brain networks exhibit increased cognitive performance, regardless of the task. Critically, we find evidence consistent with a mechanistic model in which connector hubs tune the connectivity of their neighbors to be more modular while allowing for task appropriate information integration across communities, which increases global modularity and cognitive performance.

Network Neuroscience Theory of Human Intelligence

An enduring aim of research in the psychological and brain sciences is to understand the nature of individual differences in human intelligence, examining the stunning breadth and diversity of intellectual abilities and the remarkable neurobiological mechanisms from which they arise. This Opinion article surveys recent neuroscience evidence to elucidate how general intelligence, g, emerges from individual differences in the network architecture of the human brain. The reviewed findings motivate new insights about how network topology and dynamics account for individual differences in g, represented by the Network Neuroscience Theory. According to this framework, g emerges from the small-world topology of brain networks and the dynamic reorganization of its community structure in the service of system-wide flexibility and adaptation.

The diverse club

A complex system can be represented and analyzed as a network, where nodes represent the units of the network and edges represent connections between those units. For example, a brain network represents neurons as nodes and axons between neurons as edges. In many networks, some nodes have a disproportionately high number of edges as well as many edges between each other and are referred to as the "rich club". In many different networks, the nodes of this club are assumed to support global network integration. Here we show that another set of nodes, which have edges diversely distributed across the network, form a "diverse club". The diverse club exhibits, to a greater extent than the rich club, properties consistent with an integrative network function-these nodes are more highly interconnected and their edges are more critical for efficient global integration. Finally, these two clubs potentially evolved via distinct selection pressures.

Organization and hierarchy of the human functional brain network lead to a chain-like core

The brain is a paradigmatic example of a complex system: its functionality emerges as a global property of local mesoscopic and microscopic interactions. Complex network theory allows to elicit the functional architecture of the brain in terms of links (correlations) between nodes (grey matter regions) and to extract information out of the noise. Here we present the analysis of functional magnetic resonance imaging data from forty healthy humans at rest for the investigation of the basal scaffold of the functional brain network organization. We show how brain regions tend to coordinate by forming a highly hierarchical chain-like structure of homogeneously clustered anatomical areas. A maximum spanning tree approach revealed the centrality of the occipital cortex and the peculiar aggregation of cerebellar regions to form a closed core. We also report the hierarchy of network segregation and the level of clusters integration as a function of the connectivity strength between brain regions.

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

BACKGROUND

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

NEW METHOD

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

RESULTS

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

COMPARISON WITH EXISTING METHODS

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

CONCLUSIONS

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

Synchronization patterns: from network motifs to hierarchical networks

We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks.This article is part of the themed issue 'Horizons of cybernetical physics'.

Hierarchical organization of functional connectivity in the mouse brain: a complex network approach

This paper represents a contribution to the study of the brain functional connectivity from the perspective of complex networks theory. More specifically, we apply graph theoretical analyses to provide evidence of the modular structure of the mouse brain and to shed light on its hierarchical organization. We propose a novel percolation analysis and we apply our approach to the analysis of a resting-state functional MRI data set from 41 mice. This approach reveals a robust hierarchical structure of modules persistent across different subjects. Importantly, we test this approach against a statistical benchmark (or null model) which constrains only the distributions of empirical correlations. Our results unambiguously show that the hierarchical character of the mouse brain modular structure is not trivially encoded into this lower-order constraint. Finally, we investigate the modular structure of the mouse brain by computing the Minimal Spanning Forest, a technique that identifies subnetworks characterized by the strongest internal correlations. This approach represents a faster alternative to other community detection methods and provides a means to rank modules on the basis of the strength of their internal edges.

Growth, collapse, and self-organized criticality in complex networks

Network growth is ubiquitous in nature (e.g., biological networks) and technological systems (e.g., modern infrastructures). To understand how certain dynamical behaviors can or cannot persist as the underlying network grows is a problem of increasing importance in complex dynamical systems as well as sustainability science and engineering. We address the question of whether a complex network of nonlinear oscillators can maintain its synchronization stability as it expands. We find that a large scale avalanche over the entire network can be triggered in the sense that the individual nodal dynamics diverges from the synchronous state in a cascading manner within a relatively short time period. In particular, after an initial stage of linear growth, the network typically evolves into a critical state where the addition of a single new node can cause a group of nodes to lose synchronization, leading to synchronization collapse for the entire network. A statistical analysis reveals that the collapse size is approximately algebraically distributed, indicating the emergence of self-organized criticality. We demonstrate the generality of the phenomenon of synchronization collapse using a variety of complex network models, and uncover the underlying dynamical mechanism through an eigenvector analysis.

Is functional integration of resting state brain networks an unspecific biomarker for working memory performance?

Is there one optimal topology of functional brain networks at rest from which our cognitive performance would profit? Previous studies suggest that functional integration of resting state brain networks is an important biomarker for cognitive performance. However, it is still unknown whether higher network integration is an unspecific predictor for good cognitive performance or, alternatively, whether specific network organization during rest predicts only specific cognitive abilities. Here, we investigated the relationship between network integration at rest and cognitive performance using two tasks that measured different aspects of working memory; one task assessed visual-spatial and the other numerical working memory. Network clustering, modularity and efficiency were computed to capture network integration on different levels of network organization, and to statistically compare their correlations with the performance in each working memory test. The results revealed that each working memory aspect profits from a different resting state topology, and the tests showed significantly different correlations with each of the measures of network integration. While higher global network integration and modularity predicted significantly better performance in visual-spatial working memory, both measures showed no significant correlation with numerical working memory performance. In contrast, numerical working memory was superior in subjects with highly clustered brain networks, predominantly in the intraparietal sulcus, a core brain region of the working memory network. Our findings suggest that a specific balance between local and global functional integration of resting state brain networks facilitates special aspects of cognitive performance. In the context of working memory, while visual-spatial performance is facilitated by globally integrated functional resting state brain networks, numerical working memory profits from increased capacities for local processing, especially in brain regions involved in working memory performance.

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

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

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

Neural field theory insights are used to derive effective brain connectivity matrices from the functional connectivity matrix defined by activity covariances. The symmetric case is exactly solved for a resting state system driven by white noise, in which strengths of connections, often termed effective connectivities, are inferred from functional data; these include strengths of connections that are underestimated or not detected by anatomical imaging. Proximity to criticality is calculated and found to be consistent with estimates obtainable from other methods. Links between anatomical, effective, and functional connectivity and resting state activity are quantified, with applicability to other complex networks. Proof-of-principle results are illustrated using published experimental data on anatomical connectivity and resting state functional connectivity. In particular, it is shown that functional connection matrices can be used to uncover the existence and strength of connections that are missed from anatomical connection matrices, including interhemispheric connections that are difficult to track with techniques such as diffusion spectrum imaging.

Discrete-network versus modal representations of brain activity: why a sparse regions-of-interest approach can work for analysis of continuous dynamics

The efficacy of the common practice of tracking brain dynamics using a few key regions of interest is explained via the fact that these regions are sensitive to underlying extended modes of activity, not just local dynamics. This underlines the inseparable interplay between modes and regions and reflects the reality that brain functions range from highly localized to highly extended.

Using geometry to uncover relationships between isotropy, homogeneity, and modularity in cortical connectivity

Inferences of strong modular and hierarchical structure from some cortical network studies conflict with the broadly isotropic homogeneous connectivity that has been found to a first approximation in classical anatomical studies. This conflict is resolved via consideration of the geometry of the cortex. A new geometrically based connection matrix (CM) visualization method is used to better compare experimental CMs with model CMs and thereby minimize appearance of artifacts. Model networks based on spherical geometry containing similar isotropic, homogeneous connection distributions to the experiment are shown to reproduce, interrelate, and explain key properties of experimentally derived networks, such as clustering coefficient (CC), path length, mean degree, and modularity score, using only two parameters that are fitted to an experimental spatial connectivity distribution. A greater CC in the experiment than the model indicates that, while isotropy and homogeneity of connections is a good first approximation, connections at shorter range may exhibit additional perturbations that increase clustering. These geometrically based models provide a comparative framework to assist in the next stage of revealing and analyzing anisotropic and/or inhomogeneous connections in data and their effects on network properties and visualization.

Neural masses and fields in dynamic causal modeling

Dynamic causal modeling (DCM) provides a framework for the analysis of effective connectivity among neuronal subpopulations that subtend invasive (electrocorticograms and local field potentials) and non-invasive (electroencephalography and magnetoencephalography) electrophysiological responses. This paper reviews the suite of neuronal population models including neural masses, fields and conductance-based models that are used in DCM. These models are expressed in terms of sets of differential equations that allow one to model the synaptic underpinnings of connectivity. We describe early developments using neural mass models, where convolution-based dynamics are used to generate responses in laminar-specific populations of excitatory and inhibitory cells. We show that these models, though resting on only two simple transforms, can recapitulate the characteristics of both evoked and spectral responses observed empirically. Using an identical neuronal architecture, we show that a set of conductance based models-that consider the dynamics of specific ion-channels-present a richer space of responses; owing to non-linear interactions between conductances and membrane potentials. We propose that conductance-based models may be more appropriate when spectra present with multiple resonances. Finally, we outline a third class of models, where each neuronal subpopulation is treated as a field; in other words, as a manifold on the cortical surface. By explicitly accounting for the spatial propagation of cortical activity through partial differential equations (PDEs), we show that the topology of connectivity-through local lateral interactions among cortical layers-may be inferred, even in the absence of spatially resolved data. We also show that these models allow for a detailed analysis of structure-function relationships in the cortex. Our review highlights the relationship among these models and how the hypothesis asked of empirical data suggests an appropriate model class.

Multiobjective identification of controlling areas in neuronal networks

In this paper, we investigate the multiobjective identification of controlling areas in the neuronal network of a cat's brain by considering two measures of controllability simultaneously. By utilizing nondominated sorting mechanisms and composite differential evolution (CoDE), a reference-point-based nondominated sorting composite differential evolution (RP-NSCDE) is developed to tackle the multiobjective identification of controlling areas in the neuronal network. The proposed RP-NSCDE shows its promising performance in terms of accuracy and convergence speed, in comparison to nondominated sorting genetic algorithms II. The proposed method is also compared with other representative statistical methods in the complex network theory, single objective, and constraint optimization methods to illustrate its effectiveness and reliability. It is shown that there exists a tradeoff between minimizing two objectives, and therefore pareto fronts (PFs) can be plotted. The developed approaches and findings can also be applied to coordination control of various kinds of real-world complex networks including biological networks and social networks, and so on.

Topological dynamics in spike-timing dependent plastic model neural networks

Spike-timing dependent plasticity (STDP) is a biologically constrained unsupervised form of learning that potentiates or depresses synaptic connections based on the precise timing of pre-synaptic and post-synaptic firings. The effects of on-going STDP on the topology of evolving model neural networks were assessed in 50 unique simulations which modeled 2 h of activity. After a period of stabilization, a number of global and local topological features were monitored periodically to quantify on-going changes in network structure. Global topological features included the total number of remaining synapses, average synaptic strengths, and average number of synapses per neuron (degree). Under a range of different input regimes and initial network configurations, each network maintained a robust and highly stable global structure across time. Local topology was monitored by assessing state changes of all three-neuron subgraphs (triads) present in the networks. Overall counts and the range of triad configurations varied little across the simulations; however, a substantial set of individual triads continued to undergo rapid state changes and revealed a dynamic local topology. In addition, specific small-world properties also fluctuated across time. These findings suggest that on-going STDP provides an efficient means of selecting and maintaining a stable yet flexible network organization.

Stability constraints on large-scale structural brain networks

Stability is an important dynamical property of complex systems and underpins a broad range of coherent self-organized behavior. Based on evidence that some neurological disorders correspond to linear instabilities, we hypothesize that stability constrains the brain's electrical activity and influences its structure and physiology. Using a physiologically-based model of brain electrical activity, we investigated the stability and dispersion solutions of networks of neuronal populations with propagation time delays and dendritic time constants. We find that stability is determined by the spectrum of the network's matrix of connection strengths and is independent of the temporal damping rate of axonal propagation with stability restricting the spectrum to a region in the complex plane. Time delays and dendritic time constants modify the shape of this region but it always contains the unit disk. Instabilities resulting from changes in connection strength initially have frequencies less than a critical frequency. For physiologically plausible parameter values based on the corticothalamic system, this critical frequency is approximately 10 Hz. For excitatory networks and networks with randomly distributed excitatory and inhibitory connections, time delays and non-zero dendritic time constants have no impact on network stability but do effect dispersion frequencies. Random networks with both excitatory and inhibitory connections can have multiple marginally stable modes at low delta frequencies.

Spectral characterization of hierarchical network modularity and limits of modularity detection

Many real world networks are reported to have hierarchically modular organization. However, there exists no algorithm-independent metric to characterize hierarchical modularity in a complex system. The main results of the paper are a set of methods to address this problem. First, classical results from random matrix theory are used to derive the spectrum of a typical stochastic block model hierarchical modular network form. Second, it is shown that hierarchical modularity can be fingerprinted using the spectrum of its largest eigenvalues and gaps between clusters of closely spaced eigenvalues that are well separated from the bulk distribution of eigenvalues around the origin. Third, some well-known results on fingerprinting non-hierarchical modularity in networks automatically follow as special cases, threreby unifying these previously fragmented results. Finally, using these spectral results, it is found that the limits of detection of modularity can be empirically established by studying the mean values of the largest eigenvalues and the limits of the bulk distribution of eigenvalues for an ensemble of networks. It is shown that even when modularity and hierarchical modularity are present in a weak form in the network, they are impossible to detect, because some of the leading eigenvalues fall within the bulk distribution. This provides a threshold for the detection of modularity. Eigenvalue distributions of some technological, social, and biological networks are studied, and the implications of detecting hierarchical modularity in real world networks are discussed.

Anatomical connectivity and the resting state activity of large cortical networks

This paper uses mathematical modelling and simulations to explore the dynamics that emerge in large scale cortical networks, with a particular focus on the topological properties of the structural connectivity and its relationship to functional connectivity. We exploit realistic anatomical connectivity matrices (from diffusion spectrum imaging) and investigate their capacity to generate various types of resting state activity. In particular, we study emergent patterns of activity for realistic connectivity configurations together with approximations formulated in terms of neural mass or field models. We find that homogenous connectivity matrices, of the sort of assumed in certain neural field models give rise to damped spatially periodic modes, while more localised modes reflect heterogeneous coupling topologies. When simulating resting state fluctuations under realistic connectivity, we find no evidence for a spectrum of spatially periodic patterns, even when grouping together cortical nodes into communities, using graph theory. We conclude that neural field models with translationally invariant connectivity may be best applied at the mesoscopic scale and that more general models of cortical networks that embed local neural fields, may provide appropriate models of macroscopic cortical dynamics over the whole brain.

Dynamics and processing in finite self-similar networks

A common feature of biological networks is the geometrical property of self-similarity. Molecular regulatory networks through to circulatory systems, nervous systems, social systems and ecological trophic networks show self-similar connectivity at multiple scales. We analyse the relationship between topology and signalling in contrasting classes of such topologies. We find that networks differ in their ability to contain or propagate signals between arbitrary nodes in a network depending on whether they possess branching or loop-like features. Networks also differ in how they respond to noise, such that one allows for greater integration at high noise, and this performance is reversed at low noise. Surprisingly, small-world topologies, with diameters logarithmic in system size, have slower dynamical time scales, and may be less integrated (more modular) than networks with longer path lengths. All of these phenomena are essentially mesoscopic, vanishing in the infinite limit but producing strong effects at sizes and time scales relevant to biology.

Building blocks of self-sustained activity in a simple deterministic model of excitable neural networks

Understanding the interplay of topology and dynamics of excitable neural networks is one of the major challenges in computational neuroscience. Here we employ a simple deterministic excitable model to explore how network-wide activation patterns are shaped by network architecture. Our observables are co-activation patterns, together with the average activity of the network and the periodicities in the excitation density. Our main results are: (1) the dependence of the correlation between the adjacency matrix and the instantaneous (zero time delay) co-activation matrix on global network features (clustering, modularity, scale-free degree distribution), (2) a correlation between the average activity and the amount of small cycles in the graph, and (3) a microscopic understanding of the contributions by 3-node and 4-node cycles to sustained activity.

Corticothalamic dynamics: structure of parameter space, spectra, instabilities, and reduced model

Linear instabilities are analyzed in a physiologically based mean-field corticothalamic model and a reduced-parameter model derived from it. In both models, the stable zone corresponding to normal arousal states is bounded by a series of surfaces demarcating the onsets of instabilities. The stable zone is found to depend on delay and rate parameters, whose values have a simple relationship to the number of instabilities and dominant frequencies on the stable zone's boundary. The dominant frequencies of linear activity inside the stable zone are found to lie in clearly delineated regions, each corresponding to an instability surface on its boundary and having approximately the same dominant frequency. These regions are ordered in parameter space according to their dominant frequencies, and an instability associated with the intrathalamic loop is shown to have the highest frequency that can become unstable. This reveals an important role for the thalamus in controlling the stability and bandwidth of dynamics in the corticothalamic system as a whole. The reduced model is found to agree well with the full model in a wide region of parameter space and, thus, is a useful guide to the full model's dynamics.

Interrelating anatomical, effective, and functional brain connectivity using propagators and neural field theory

It is shown how to compute effective and functional connection matrices (eCMs and fCMs) from anatomical CMs (aCMs) and corresponding strength-of-connection matrices (sCMs) using propagator methods in which neural interactions play the role of scatterings. This analysis demonstrates how network effects dress the bare propagators (the sCMs) to yield effective propagators (the eCMs) that can be used to compute the covariances customarily used to define fCMs. The results incorporate excitatory and inhibitory connections, multiple structures and populations, asymmetries, time delays, and measurement effects. They can also be postprocessed in the same manner as experimental measurements for direct comparison with data and thereby give insights into the role of coarse-graining, thresholding, and other effects in determining the structure of CMs. The spatiotemporal results show how to generalize CMs to include time delays and how natural network modes give rise to long-range coherence at resonant frequencies. The results are demonstrated using tractable analytic cases via neural field theory of cortical and corticothalamic systems. These also demonstrate close connections between the structure of CMs and proximity to critical points of the system, highlight the importance of indirect links between brain regions and raise the possibility of imaging specific levels of indirect connectivity. Aside from the results presented explicitly here, the expression of the connections among aCMs, sCMs, eCMs, and fCMs in terms of propagators opens the way for propagator theory to be further applied to analysis of connectivity.

Brain graphs: graphical models of the human brain connectome

Brain graphs provide a relatively simple and increasingly popular way of modeling the human brain connectome, using graph theory to abstractly define a nervous system as a set of nodes (denoting anatomical regions or recording electrodes) and interconnecting edges (denoting structural or functional connections). Topological and geometrical properties of these graphs can be measured and compared to random graphs and to graphs derived from other neuroscience data or other (nonneural) complex systems. Both structural and functional human brain graphs have consistently demonstrated key topological properties such as small-worldness, modularity, and heterogeneous degree distributions. Brain graphs are also physically embedded so as to nearly minimize wiring cost, a key geometric property. Here we offer a conceptual review and methodological guide to graphical analysis of human neuroimaging data, with an emphasis on some of the key assumptions, issues, and trade-offs facing the investigator.

Geometric effects on complex network structure in the cortex

It is shown that homogeneous, short-range, two-dimensional (2D) cortical connectivity, without modularity, hierarchy, or other specialized structure, reproduces key observed properties of cortical networks, including low path length, high clustering and modularity index, and apparent hierarchical block-diagonal structure in connection matrices. Geometry strongly influences connection matrices, implying that simple interpretations of connectivity measures as reflecting specialized structure can be misleading: Such apparent structure is seen in strictly uniform, locally connected architectures in 2D. Geometry is thus a proxy for function, modularity, and hierarchy and must be accounted for when structural inferences are made.

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.

Network growth under the constraint of synchronization stability

While it is well recognized that realistic networks are typically growing with time, the dynamical features of their growing processes remain to be explored. In the present paper, incorporating the requirement of synchronization stability into the conventional models of network growth, we will investigate how the growing process of a complex network is influenced by, and also will influence, the network collective dynamics. Our study shows that, constrained by the synchronization stability, the network will be growing in a selective and dynamical fashion. In particular, we find that the chance for a new node to be accepted by the growing network could have a large variation, i.e., it follows roughly a power-law distribution. Furthermore, we find that, with the dynamical growth, the network is always developed into structures of clear scale-free features, despite the form of the link attachment (preferential or random). The dynamical properties of network growth are studied using the method of eigenvalue analysis, and they are verified by direct simulations of coupled chaotic oscillators. Our study implies that, driven by the network collective dynamics, network growth could also be highly dynamical.

Age-related changes in topological organization of structural brain networks in healthy individuals

The aim of this study was to examine structural brain networks using regional gray matter volume, as well as to investigate changes in small-world and modular organization with normal aging. We constructed structural brain networks composed of 90 regions in young, middle, and old age groups. We randomly selected 350 healthy subjects for each group from a Japanese magnetic resonance image database. Structural brain networks in three age groups showed economical small-world properties, providing high global and local efficiency for parallel information processing at low connection cost. The small-world efficiency and node betweenness varied significantly and revealed a U- or inverted U-curve model tendency among three age groups. Results also demonstrated that structural brain networks exhibited a modular organization in which the connections between regions are much denser within modules than between them. The modular organization of structural brain networks was similar between the young and middle age groups, but quite different from the old group. In particular, the old group showed a notable decrease in the connector ratio and the intermodule connections. Combining the results of small-world efficiency, node betweenness and modular organization, we concluded that the brain network changed slightly, developing into a more distributed organization from young to middle age. The organization eventually altered greatly, shifting to a more localized organization in old age. Our findings provided quantitative insights into topological principles of structural brain networks and changes related to normal aging.

Altered small-world brain networks in temporal lobe in patients with schizophrenia performing an auditory oddball task

The functional architecture of the human brain has been extensively described in terms of complex networks characterized by efficient small-world features. Recent functional magnetic resonance imaging (fMRI) studies have found altered small-world topological properties of brain functional networks in patients with schizophrenia (SZ) during the resting state. However, little is known about the small-world properties of brain networks in the context of a task. In this study, we investigated the topological properties of human brain functional networks derived from fMRI during an auditory oddball (AOD) task. Data were obtained from 20 healthy controls and 20 SZ; A left and a right task-related network which consisted of the top activated voxels in temporal lobe of each hemisphere were analyzed separately. All voxels were detected by group independent component analysis. Connectivity of the left and right task-related networks were estimated by partial correlation analysis and thresholded to construct a set of undirected graphs. The small-worldness values were decreased in both hemispheres in SZ. In addition, SZ showed longer shortest path length and lower global efficiency only in the left task-related networks. These results suggested small-world attributes are altered during the AOD task-related networks in SZ which provided further evidences for brain dysfunction of connectivity in SZ.

Modular and hierarchically modular organization of brain networks

Brain networks are increasingly understood as one of a large class of information processing systems that share important organizational principles in common, including the property of a modular community structure. A module is topologically defined as a subset of highly inter-connected nodes which are relatively sparsely connected to nodes in other modules. In brain networks, topological modules are often made up of anatomically neighboring and/or functionally related cortical regions, and inter-modular connections tend to be relatively long distance. Moreover, brain networks and many other complex systems demonstrate the property of hierarchical modularity, or modularity on several topological scales: within each module there will be a set of sub-modules, and within each sub-module a set of sub-sub-modules, etc. There are several general advantages to modular and hierarchically modular network organization, including greater robustness, adaptivity, and evolvability of network function. In this context, we review some of the mathematical concepts available for quantitative analysis of (hierarchical) modularity in brain networks and we summarize some of the recent work investigating modularity of structural and functional brain networks derived from analysis of human neuroimaging data.

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

Normal functioning in many realistic complex dynamical systems, such as neural networks, requires coherence and synchronization for collective actions of network components. However, strong synchronization of the whole network is often related to pathological situations. A regime in between enabling both segregation in subsystems and integration as a whole is thus desirable. Here, we characterize this regime by complexity of synchronization patterns and study its relationship to heterogeneous and modular architecture in complex network of oscillators. We show that these networks possess a broad range of high complexity associated with the formation of dynamical clusters and the coordination between the clusters. In realistic networks of C. elegans and cat cortex, the complexity is reduced when the original network is rewired in various ways, reflecting that the neural systems are organized to provide a combination of segregation and integration with the coexistence of various complex network features, especially modularity and heterogeneity. Our work can stimulate further studies on structure-function relationships in neural systems through the inquiry of the specific functional roles of the intermediate dynamical regime.

Optimal hierarchical modular topologies for producing limited sustained activation of neural networks

An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable regimes of network activation, typically arising from a limited parameter range. In this range of limited sustained activity (LSA), the activity of neural populations in the network persists between the extremes of either quickly dying out or activating the whole network. Hierarchical modular networks were previously found to show a wider parameter range for LSA than random or small-world networks not possessing hierarchical organization or multiple modules. Here we explored how variation in the number of hierarchical levels and modules per level influenced network dynamics and occurrence of LSA. We tested hierarchical configurations of different network sizes, approximating the large-scale networks linking cortical columns in one hemisphere of the rat, cat, or macaque monkey brain. Scaling of the network size affected the number of hierarchical levels and modules in the optimal networks, also depending on whether global edge density or the numbers of connections per node were kept constant. For constant edge density, only few network configurations, possessing an intermediate number of levels and a large number of modules, led to a large range of LSA independent of brain size. For a constant number of node connections, there was a trend for optimal configurations in larger-size networks to possess a larger number of hierarchical levels or more modules. These results may help to explain the trend to greater network complexity apparent in larger brains and may indicate that this complexity is required for maintaining stable levels of neural activation.

Efficient physical embedding of topologically complex information processing networks in brains and computer circuits

Nervous systems are information processing networks that evolved by natural selection, whereas very large scale integrated (VLSI) computer circuits have evolved by commercially driven technology development. Here we follow historic intuition that all physical information processing systems will share key organizational properties, such as modularity, that generally confer adaptivity of function. It has long been observed that modular VLSI circuits demonstrate an isometric scaling relationship between the number of processing elements and the number of connections, known as Rent's rule, which is related to the dimensionality of the circuit's interconnect topology and its logical capacity. We show that human brain structural networks, and the nervous system of the nematode C. elegans, also obey Rent's rule, and exhibit some degree of hierarchical modularity. We further show that the estimated Rent exponent of human brain networks, derived from MRI data, can explain the allometric scaling relations between gray and white matter volumes across a wide range of mammalian species, again suggesting that these principles of nervous system design are highly conserved. For each of these fractal modular networks, the dimensionality of the interconnect topology was greater than the 2 or 3 Euclidean dimensions of the space in which it was embedded. This relatively high complexity entailed extra cost in physical wiring: although all networks were economically or cost-efficiently wired they did not strictly minimize wiring costs. Artificial and biological information processing systems both may evolve to optimize a trade-off between physical cost and topological complexity, resulting in the emergence of homologous principles of economical, fractal and modular design across many different kinds of nervous and computational networks.

Brains are often compared to computers but, apart from the trivial fact that both process information using a complex physical pattern of connections, it has been unclear whether this is more than just a metaphor. In our work, we rigorously uncover novel quantitative organizational principles that underlie the network organization of the human brain, high performance computer circuits, and the nervous system of the nematode C. elegans. We show through a topological and physical analysis of connectivity data that each of these systems is cost-efficiently embedded in physical space; they are organized as economical modular networks, paying a modest premium in wiring cost for the functional advantages of high dimensional topology. We also show that the fractal properties of human brain network connectivity can be used to explain allometric scaling relations between grey and white matter volumes in the brains of a wide range of differently sized mammals—from mouse opossum to sea lion—further suggesting that these principles of nervous system design are highly conserved across species. We propose that market-driven human invention and natural selection have negotiated trade-offs between cost and complexity in design of information processing networks and convergently come to similar conclusions.