Metastable chimera states in community-structured oscillator networks

Imperfectly synchronized states and chimera states in two interacting populations of nonlocally coupled Stuart-Landau oscillators

We investigate the emergence of different kinds of imperfectly synchronized states and chimera states in two interacting populations of nonlocally coupled Stuart-Landau oscillators. We find that the complete synchronization in population I and existence of solitary oscillators which escape from the synchronized group in population II lead to imperfectly synchronized states for sufficiently small values of nonisochronicity parameter. Interestingly, upon increasing the strength of this parameter further there occurs an onset of mixed imperfectly synchronized states where the solitary oscillators occur from both the populations. Synchronized oscillators from both the populations are locked to a common average frequency. In both cases of imperfectly synchronized states, synchronized oscillators exhibit periodic motion while the solitary oscillators are quasiperiodic in nature. In this region, for spatially prepared initial conditions, we can observe the mixed chimera states where the coexistence of synchronized and desynchronized oscillations occur from both the populations. On the other hand, imperfectly synchronized states are not always stable, and they can drift aperiodically due to instability caused by an increase of nonisochronicity parameter. We observe that these states are robust to the introduction of frequency mismatch between the two populations.

Modularity Induced Gating and Delays in Neuronal Networks

Neural networks, despite their highly interconnected nature, exhibit distinctly localized and gated activation. Modularity, a distinctive feature of neural networks, has been recently proposed as an important parameter determining the manner by which networks support activity propagation. Here we use an engineered biological model, consisting of engineered rat cortical neurons, to study the role of modular topology in gating the activity between cell populations. We show that pairs of connected modules support conditional propagation (transmitting stronger bursts with higher probability), long delays and propagation asymmetry. Moreover, large modular networks manifest diverse patterns of both local and global activation. Blocking inhibition decreased activity diversity and replaced it with highly consistent transmission patterns. By independently controlling modularity and disinhibition, experimentally and in a model, we pose that modular topology is an important parameter affecting activation localization and is instrumental for population-level gating by disinhibition.

The capacity to transmit information between connected parts of a neuronal network is fundamental to its function. The organization of network connections (the topology of the network) is therefore expected to play an important role in determining network transmission. Since modular topology characterizes many brain circuits on multiple scales, investigating the role of modularity in activity gating is clearly desirable. By engineering such modular networks in vitro, we were able to perform such an investigation. Under these experimental conditions, we can independently control the degree of modularity, as well as inhibition in the network. We show that a combination of these two properties is highly beneficial from a communication perspective. Namely, it equips connected modules and large modular networks with the capacity to gate and temporally coordinate activity between the different parts of the network.

Dynamic competition between large-scale functional networks differentiates fear conditioning and extinction in humans

The high evolutionary value of learning when to respond to threats or when to inhibit previously learned associations after changing threat contingencies is reflected in dedicated networks in the animal and human brain. Recent evidence further suggests that adaptive learning may be dependent on the dynamic interaction of meta-stable functional brain networks. However, it is still unclear which functional brain networks compete with each other to facilitate associative learning and how changes in threat contingencies affect this competition. The aim of this study was to assess the dynamic competition between large-scale networks related to associative learning in the human brain by combining a repeated differential conditioning and extinction paradigm with independent component analysis of functional magnetic resonance imaging data. The results (i) identify three task-related networks involved in initial and sustained conditioning as well as extinction, and demonstrate that (ii) the two main networks that underlie sustained conditioning and extinction are anti-correlated with each other and (iii) the dynamic competition between these two networks is modulated in response to changes in associative contingencies. These findings provide novel evidence for the view that dynamic competition between large-scale functional networks differentiates fear conditioning from extinction learning in the healthy brain and suggest that dysfunctional network dynamics might contribute to learning-related neuropsychiatric disorders.

Neurocomputational Model of EEG Complexity during Mind Wandering

Mind wandering (MW) can be understood as a transient state in which attention drifts from an external task to internal self-generated thoughts. MW has been associated with the activation of the Default Mode Network (DMN). In addition, it has been shown that the activity of the DMN is anti-correlated with activation in brain networks related to the processing of external events (e.g., Salience network, SN). In this study, we present a mean field model based on weakly coupled Kuramoto oscillators. We simulated the oscillatory activity of the entire brain and explored the role of the interaction between the nodes from the DMN and SN in MW states. External stimulation was added to the network model in two opposite conditions. Stimuli could be presented when oscillators in the SN showed more internal coherence (synchrony) than in the DMN, or, on the contrary, when the coherence in the SN was lower than in the DMN. The resulting phases of the oscillators were analyzed and used to simulate EEG signals. Our results showed that the structural complexity from both simulated and real data was higher when the model was stimulated during periods in which DMN was more coherent than the SN. Overall, our results provided a plausible mechanistic explanation to MW as a state in which high coherence in the DMN partially suppresses the capacity of the system to process external stimuli.

Chimera-like States in Modular Neural Networks

Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider a neural network inspired by the connectome of the C. elegans soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter ρ, we also employ other measures of coherence, such as the chimera-like χ and metastability λ indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the C. elegans neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks.

Phase oscillators in modular networks: The effect of nonlocal coupling

We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the different dynamical states.

Criticality as a Set-Point for Adaptive Behavior in Neuromorphic Hardware

Neuromorphic hardware are designed by drawing inspiration from biology to overcome limitations of current computer architectures while forging the development of a new class of autonomous systems that can exhibit adaptive behaviors. Several designs in the recent past are capable of emulating large scale networks but avoid complexity in network dynamics by minimizing the number of dynamic variables that are supported and tunable in hardware. We believe that this is due to the lack of a clear understanding of how to design self-tuning complex systems. It has been widely demonstrated that criticality appears to be the default state of the brain and manifests in the form of spontaneous scale-invariant cascades of neural activity. Experiment, theory and recent models have shown that neuronal networks at criticality demonstrate optimal information transfer, learning and information processing capabilities that affect behavior. In this perspective article, we argue that understanding how large scale neuromorphic electronics can be designed to enable emergent adaptive behavior will require an understanding of how networks emulated by such hardware can self-tune local parameters to maintain criticality as a set-point. We believe that such capability will enable the design of truly scalable intelligent systems using neuromorphic hardware that embrace complexity in network dynamics rather than avoiding it.

Approximating frustration scores in complex networks via perturbed Laplacian spectra

Systems of many interacting components, as found in physics, biology, infrastructure, and the social sciences, are often modeled by simple networks of nodes and edges. The real-world systems frequently confront outside intervention or internal damage whose impact must be predicted or minimized, and such perturbations are then mimicked in the models by altering nodes or edges. This leads to the broad issue of how to best quantify changes in a model network after some type of perturbation. In the case of node removal there are many centrality metrics which associate a scalar quantity with the removed node, but it can be difficult to associate the quantities with some intuitive aspect of physical behavior in the network. This presents a serious hurdle to the application of network theory: real-world utility networks are rarely altered according to theoretic principles unless the kinetic impact on the network's users are fully appreciated beforehand. In pursuit of a kinetically interpretable centrality score, we discuss the f-score, or frustration score. Each f-score quantifies whether a selected node accelerates or inhibits global mean first passage times to a second, independently selected target node. We show that this is a natural way of revealing the dynamical importance of a node in some networks. After discussing merits of the f-score metric, we combine spectral and Laplacian matrix theory in order to quickly approximate the exact f-score values, which can otherwise be expensive to compute. Following tests on both synthetic and real medium-sized networks, we report f-score runtime improvements over exact brute force approaches in the range of 0 to 400% with low error (<3%).

Local inhibitory plasticity tunes macroscopic brain dynamics and allows the emergence of functional brain networks

Rich, spontaneous brain activity has been observed across a range of different temporal and spatial scales. These dynamics are thought to be important for efficient neural functioning. A range of experimental evidence suggests that these neural dynamics are maintained across a variety of different cognitive states, in response to alterations of the environment and to changes in brain configuration (e.g., across individuals, development and in many neurological disorders). This suggests that the brain has evolved mechanisms to maintain rich dynamics across a broad range of situations. Several mechanisms based around homeostatic plasticity have been proposed to explain how these dynamics emerge from networks of neurons at the microscopic scale. Here we explore how a homeostatic mechanism may operate at the macroscopic scale: in particular, focusing on how it interacts with the underlying structural network topology and how it gives rise to well-described functional connectivity networks. We use a simple mean-field model of the brain, constrained by empirical white matter structural connectivity where each region of the brain is simulated using a pool of excitatory and inhibitory neurons. We show, as with the microscopic work, that homeostatic plasticity regulates network activity and allows for the emergence of rich, spontaneous dynamics across a range of brain configurations, which otherwise show a very limited range of dynamic regimes. In addition, the simulated functional connectivity of the homeostatic model better resembles empirical functional connectivity network. To accomplish this, we show how the inhibitory weights adapt over time to capture important graph theoretic properties of the underlying structural network. Therefore, this work presents suggests how inhibitory homeostatic mechanisms facilitate stable macroscopic dynamics to emerge in the brain, aiding the formation of functional connectivity networks.

Cognitive Flexibility through Metastable Neural Dynamics Is Disrupted by Damage to the Structural Connectome

Current theory proposes that healthy neural dynamics operate in a metastable regime, where brain regions interact to simultaneously maximize integration and segregation. Metastability may confer important behavioral properties, such as cognitive flexibility. It is increasingly recognized that neural dynamics are constrained by the underlying structural connections between brain regions. An important challenge is, therefore, to relate structural connectivity, neural dynamics, and behavior. Traumatic brain injury (TBI) is a pre-eminent structural disconnection disorder whereby traumatic axonal injury damages large-scale connectivity, producing characteristic cognitive impairments, including slowed information processing speed and reduced cognitive flexibility, that may be a result of disrupted metastable dynamics. Therefore, TBI provides an experimental and theoretical model to examine how metastable dynamics relate to structural connectivity and cognition. Here, we use complementary empirical and computational approaches to investigate how metastability arises from the healthy structural connectome and relates to cognitive performance. We found reduced metastability in large-scale neural dynamics after TBI, measured with resting-state functional MRI. This reduction in metastability was associated with damage to the connectome, measured using diffusion MRI. Furthermore, decreased metastability was associated with reduced cognitive flexibility and information processing. A computational model, defined by empirically derived connectivity data, demonstrates how behaviorally relevant changes in neural dynamics result from structural disconnection. Our findings suggest how metastable dynamics are important for normal brain function and contingent on the structure of the human connectome.

Complexity of Multi-Dimensional Spontaneous EEG Decreases during Propofol Induced General Anaesthesia

Emerging neural theories of consciousness suggest a correlation between a specific type of neural dynamical complexity and the level of consciousness: When awake and aware, causal interactions between brain regions are both integrated (all regions are to a certain extent connected) and differentiated (there is inhomogeneity and variety in the interactions). In support of this, recent work by Casali et al (2013) has shown that Lempel-Ziv complexity correlates strongly with conscious level, when computed on the EEG response to transcranial magnetic stimulation. Here we investigated complexity of spontaneous high-density EEG data during propofol-induced general anaesthesia. We consider three distinct measures: (i) Lempel-Ziv complexity, which is derived from how compressible the data are; (ii) amplitude coalition entropy, which measures the variability in the constitution of the set of active channels; and (iii) the novel synchrony coalition entropy (SCE), which measures the variability in the constitution of the set of synchronous channels. After some simulations on Kuramoto oscillator models which demonstrate that these measures capture distinct 'flavours' of complexity, we show that there is a robustly measurable decrease in the complexity of spontaneous EEG during general anaesthesia.

Nonlinearity of local dynamics promotes multi-chimeras

Chimera states are complex spatio-temporal patterns in which domains of synchronous and asynchronous dynamics coexist in coupled systems of oscillators. We examine how the character of the individual elements influences chimera states by studying networks of nonlocally coupled Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and demonstrate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains. We show that the stability regimes for multi-chimera states and the mean phase velocity profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore, we reveal the influence of time delay on chimera patterns.

Effects of lesions on synchrony and metastability in cortical networks

At the macroscopic scale, the human brain can be described as a complex network of white matter tracts integrating grey matter assemblies - the human connectome. The structure of the connectome, which is often described using graph theoretic approaches, can be used to model macroscopic brain function at low computational cost. Here, we use the Kuramoto model of coupled oscillators with time-delays, calibrated with respect to empirical functional MRI data, to study the relation between the structure of the connectome and two aspects of functional brain dynamics - synchrony, a measure of general coherence, and metastability, a measure of dynamical flexibility. Specifically, we investigate the relationship between the local structure of the connectome, quantified using graph theory, and the synchrony and metastability of the model's dynamics. By removing individual nodes and all of their connections from the model, we study the effect of lesions on both global and local dynamics. Of the nine nodal graph-theoretical properties tested, two were able to predict effects of node lesion on the global dynamics. The removal of nodes with high eigenvector centrality leads to decreases in global synchrony and increases in global metastability, as does the removal of hub nodes joining topologically segregated network modules. At the level of local dynamics in the neighbourhood of the lesioned node, structural properties of the lesioned nodes hold more predictive power, as five nodal graph theoretical measures are related to changes in local dynamics following node lesions. We discuss these results in the context of empirical studies of stroke and functional brain dynamics.

Cortical activity is more stable when sensory stimuli are consciously perceived

According to recent evidence, stimulus-tuned neurons in the cerebral cortex exhibit reduced variability in firing rate across trials, after the onset of a stimulus. However, in order for a reduction in variability to be directly relevant to perception and behavior, it must be realized within trial--the pattern of activity must be relatively stable. Stability is characteristic of decision states in recurrent attractor networks, and its possible relevance to conscious perception has been suggested by theorists. However, it is difficult to measure on the within-trial time scales and broadly distributed spatial scales relevant to perception. We recorded simultaneous magneto- and electroencephalography (MEG and EEG) data while subjects observed threshold-level visual stimuli. Pattern-similarity analyses applied to the data from MEG gradiometers uncovered a pronounced decrease in variability across trials after stimulus onset, consistent with previous single-unit data. This was followed by a significant divergence in variability depending upon subjective report (seen/unseen), with seen trials exhibiting less variability. Applying the same analysis across time, within trial, we found that the latter effect coincided in time with a difference in the stability of the pattern of activity. Stability alone could be used to classify data from individual trials as "seen" or "unseen." The same metric applied to EEG data from patients with disorders of consciousness exposed to auditory stimuli diverged parametrically according to clinically diagnosed level of consciousness. Differences in signal strength could not account for these results. Conscious perception may involve the transient stabilization of distributed cortical networks, corresponding to a global brain-scale decision.

Chimera states on the surface of a sphere

A chimera state is a spatiotemporal pattern in which a network of identical coupled oscillators exhibits coexisting regions of asynchronous and synchronous oscillation. Two distinct classes of chimera states have been shown to exist: "spots" and "spirals." Here we study coupled oscillators on the surface of a sphere, a single system in which both spot and spiral chimera states appear. We present an analysis of the birth and death of spiral chimera states and show that although they coexist with spot chimeras, they are stable in disjoint regions of parameter space.

Robustness of chimera states for coupled FitzHugh-Nagumo oscillators

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

Dynamic changes in network synchrony reveal resting-state functional networks

Experimental functional magnetic resonance imaging studies have shown that spontaneous brain activity, i.e., in the absence of any external input, exhibit complex spatial and temporal patterns of co-activity between segregated brain regions. These so-called large-scale resting-state functional connectivity networks represent dynamically organized neural assemblies interacting with each other in a complex way. It has been suggested that looking at the dynamical properties of complex patterns of brain functional co-activity may reveal neural mechanisms underlying the dynamic changes in functional interactions. Here, we examine how global network dynamics is shaped by different network configurations, derived from realistic brain functional interactions. We focus on two main dynamics measures: synchrony and variations in synchrony. Neural activity and the inferred hemodynamic response of the network nodes are simulated using a system of 90 FitzHugh-Nagumo neural models subject to system noise and time-delayed interactions. These models are embedded into the topology of the complex brain functional interactions, whose architecture is additionally reduced to its main structural pathways. In the simulated functional networks, patterns of correlated regional activity clearly arise from dynamical properties that maximize synchrony and variations in synchrony. Our results on the fast changes of the level of the network synchrony also show how flexible changes in the large-scale network dynamics could be.

Frustrated hierarchical synchronization and emergent complexity in the human connectome network

The spontaneous emergence of coherent behavior through synchronization plays a key role in neural function, and its anomalies often lie at the basis of pathologies. Here we employ a parsimonious (mesoscopic) approach to study analytically and computationally the synchronization (Kuramoto) dynamics on the actual human-brain connectome network. We elucidate the existence of a so-far-uncovered intermediate phase, placed between the standard synchronous and asynchronous phases, i.e. between order and disorder. This novel phase stems from the hierarchical modular organization of the connectome. Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology. We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access –in a robust though flexible way– a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point.

Structure-function discrepancy: inhomogeneity and delays in synchronized neural networks

The discrepancy between structural and functional connectivity in neural systems forms the challenge in understanding general brain functioning. To pinpoint a mapping between structure and function, we investigated the effects of (in)homogeneity in coupling structure and delays on synchronization behavior in networks of oscillatory neural masses by deriving the phase dynamics of these generic networks. For homogeneous delays, the structural coupling matrix is largely preserved in the coupling between phases, resulting in clustered stationary phase distributions. Accordingly, we found only a small number of synchronized groups in the network. Distributed delays, by contrast, introduce inhomogeneity in the phase coupling so that clustered stationary phase distributions no longer exist. The effect of distributed delays mimicked that of structural inhomogeneity. Hence, we argue that phase (de-)synchronization patterns caused by inhomogeneous coupling cannot be distinguished from those caused by distributed delays, at least not by the naked eye. The here-derived analytical expression for the effective coupling between phases as a function of structural coupling constitutes a direct relationship between structural and functional connectivity. Structural connectivity constrains synchronizability that may be modified by the delay distribution. This explains why structural and functional connectivity bear much resemblance albeit not a one-to-one correspondence. We illustrate this in the context of resting-state activity, using the anatomical connectivity structure reported by Hagmann and others.

Influence of wiring cost on the large-scale architecture of human cortical connectivity

In the past two decades some fundamental properties of cortical connectivity have been discovered: small-world structure, pronounced hierarchical and modular organisation, and strong core and rich-club structures. A common assumption when interpreting results of this kind is that the observed structural properties are present to enable the brain's function. However, the brain is also embedded into the limited space of the skull and its wiring has associated developmental and metabolic costs. These basic physical and economic aspects place separate, often conflicting, constraints on the brain's connectivity, which must be characterized in order to understand the true relationship between brain structure and function. To address this challenge, here we ask which, and to what extent, aspects of the structural organisation of the brain are conserved if we preserve specific spatial and topological properties of the brain but otherwise randomise its connectivity. We perform a comparative analysis of a connectivity map of the cortical connectome both on high- and low-resolutions utilising three different types of surrogate networks: spatially unconstrained ('random'), connection length preserving ('spatial'), and connection length optimised ('reduced') surrogates. We find that unconstrained randomisation markedly diminishes all investigated architectural properties of cortical connectivity. By contrast, spatial and reduced surrogates largely preserve most properties and, interestingly, often more so in the reduced surrogates. Specifically, our results suggest that the cortical network is less tightly integrated than its spatial constraints would allow, but more strongly segregated than its spatial constraints would necessitate. We additionally find that hierarchical organisation and rich-club structure of the cortical connectivity are largely preserved in spatial and reduced surrogates and hence may be partially attributable to cortical wiring constraints. In contrast, the high modularity and strong s-core of the high-resolution cortical network are significantly stronger than in the surrogates, underlining their potential functional relevance in the brain.

The control of global brain dynamics: opposing actions of frontoparietal control and default mode networks on attention

Understanding how dynamic changes in brain activity control behavior is a major challenge of cognitive neuroscience. Here, we consider the brain as a complex dynamic system and define two measures of brain dynamics: the synchrony of brain activity, measured by the spatial coherence of the BOLD signal across regions of the brain; and metastability, which we define as the extent to which synchrony varies over time. We investigate the relationship among brain network activity, metastability, and cognitive state in humans, testing the hypothesis that global metastability is "tuned" by network interactions. We study the following two conditions: (1) an attentionally demanding choice reaction time task (CRT); and (2) an unconstrained "rest" state. Functional MRI demonstrated increased synchrony, and decreased metastability was associated with increased activity within the frontoparietal control/dorsal attention network (FPCN/DAN) activity and decreased default mode network (DMN) activity during the CRT compared with rest. Using a computational model of neural dynamics that is constrained by white matter structure to test whether simulated changes in FPCN/DAN and DMN activity produce similar effects, we demonstate that activation of the FPCN/DAN increases global synchrony and decreases metastability. DMN activation had the opposite effects. These results suggest that the balance of activity in the FPCN/DAN and DMN might control global metastability, providing a mechanistic explanation of how attentional state is shifted between an unfocused/exploratory mode characterized by high metastability, and a focused/constrained mode characterized by low metastability.

The metastable brain

Neural ensembles oscillate across a broad range of frequencies and are transiently coupled or "bound" together when people attend to a stimulus, perceive, think, and act. This is a dynamic, self-assembling process, with parts of the brain engaging and disengaging in time. But how is it done? The theory of Coordination Dynamics proposes a mechanism called metastability, a subtle blend of integration and segregation. Tendencies for brain regions to express their individual autonomy and specialized functions (segregation, modularity) coexist with tendencies to couple and coordinate globally for multiple functions (integration). Although metastability has garnered increasing attention, it has yet to be demonstrated and treated within a fully spatiotemporal perspective. Here, we illustrate metastability in continuous neural and behavioral recordings, and we discuss theory and experiments at multiple scales, suggesting that metastable dynamics underlie the real-time coordination necessary for the brain's dynamic cognitive, behavioral, and social functions.

The role of the posterior cingulate cortex in cognition and disease

The posterior cingulate cortex is a highly connected and metabolically active brain region. Recent studies suggest it has an important cognitive role, although there is no consensus about what this is. The region is typically discussed as having a unitary function because of a common pattern of relative deactivation observed during attentionally demanding tasks. One influential hypothesis is that the posterior cingulate cortex has a central role in supporting internally-directed cognition. It is a key node in the default mode network and shows increased activity when individuals retrieve autobiographical memories or plan for the future, as well as during unconstrained 'rest' when activity in the brain is 'free-wheeling'. However, other evidence suggests that the region is highly heterogeneous and may play a direct role in regulating the focus of attention. In addition, its activity varies with arousal state and its interactions with other brain networks may be important for conscious awareness. Understanding posterior cingulate cortex function is likely to be of clinical importance. It is well protected against ischaemic stroke, and so there is relatively little neuropsychological data about the consequences of focal lesions. However, in other conditions abnormalities in the region are clearly linked to disease. For example, amyloid deposition and reduced metabolism is seen early in Alzheimer's disease. Functional neuroimaging studies show abnormalities in a range of neurological and psychiatric disorders including Alzheimer's disease, schizophrenia, autism, depression and attention deficit hyperactivity disorder, as well as ageing. Our own work has consistently shown abnormal posterior cingulate cortex function following traumatic brain injury, which predicts attentional impairments. Here we review the anatomy and physiology of the region and how it is affected in a range of clinical conditions, before discussing its proposed functions. We synthesize key findings into a novel model of the region's function (the 'Arousal, Balance and Breadth of Attention' model). Dorsal and ventral subcomponents are functionally separated and differences in regional activity are explained by considering: (i) arousal state; (ii) whether attention is focused internally or externally; and (iii) the breadth of attentional focus. The predictions of the model can be tested within the framework of complex dynamic systems theory, and we propose that the dorsal posterior cingulate cortex influences attentional focus by 'tuning' whole-brain metastability and so adjusts how stable brain network activity is over time.

Remote synchronization reveals network symmetries and functional modules

We study a Kuramoto model in which the oscillators are associated with the nodes of a complex network and the interactions include a phase frustration, thus preventing full synchronization. The system organizes into a regime of remote synchronization where pairs of nodes with the same network symmetry are fully synchronized, despite their distance on the graph. We provide analytical arguments to explain this result, and we show how the frustration parameter affects the distribution of phases. An application to brain networks suggests that anatomical symmetry plays a role in neural synchronization by determining correlated functional modules across distant locations.

Metastability and inter-band frequency modulation in networks of oscillating spiking neuron populations

Groups of neurons firing synchronously are hypothesized to underlie many cognitive functions such as attention, associative learning, memory, and sensory selection. Recent theories suggest that transient periods of synchronization and desynchronization provide a mechanism for dynamically integrating and forming coalitions of functionally related neural areas, and that at these times conditions are optimal for information transfer. Oscillating neural populations display a great amount of spectral complexity, with several rhythms temporally coexisting in different structures and interacting with each other. This paper explores inter-band frequency modulation between neural oscillators using models of quadratic integrate-and-fire neurons and Hodgkin-Huxley neurons. We vary the structural connectivity in a network of neural oscillators, assess the spectral complexity, and correlate the inter-band frequency modulation. We contrast this correlation against measures of metastable coalition entropy and synchrony. Our results show that oscillations in different neural populations modulate each other so as to change frequency, and that the interaction of these fluctuating frequencies in the network as a whole is able to drive different neural populations towards episodes of synchrony. Further to this, we locate an area in the connectivity space in which the system directs itself in this way so as to explore a large repertoire of synchronous coalitions. We suggest that such dynamics facilitate versatile exploration, integration, and communication between functionally related neural areas, and thereby supports sophisticated cognitive processing in the brain.

The brain's connective core and its role in animal cognition

This paper addresses the question of how the brain of an animal achieves cognitive integration--that is to say how it manages to bring its fullest resources to bear on an ongoing situation. To fully exploit its cognitive resources, whether inherited or acquired through experience, it must be possible for unanticipated coalitions of brain processes to form. This facilitates the novel recombination of the elements of an existing behavioural repertoire, and thereby enables innovation. But in a system comprising massively many anatomically distributed assemblies of neurons, it is far from clear how such open-ended coalition formation is possible. The present paper draws on contemporary findings in brain connectivity and neurodynamics, as well as the literature of artificial intelligence, to outline a possible answer in terms of the brain's most richly connected and topologically central structures, its so-called connective core.

Outline of a general theory of behavior and brain coordination

Much evidence suggests that dynamic laws of neurobehavioral coordination are sui generis: they deal with collective properties that are repeatable from one system to another and emerge from microscopic dynamics but may not (even in principle) be deducible from them. Nevertheless, it is useful to try to understand the relationship between different levels while all the time respecting the autonomy of each. We report a program of research that uses the theoretical concepts of coordination dynamics and quantitative measurements of simple, well-defined experimental model systems to explicitly relate neural and behavioral levels of description in human beings. Our approach is both top-down and bottom-up and aims at ending up in the same place: top-down to derive behavioral patterns from neural fields, and bottom-up to generate neural field patterns from bidirectional coupling between astrocytes and neurons. Much progress can be made by recognizing that the two approaches--reductionism and emergentism--are complementary. A key to understanding is to couch the coordination of very different things--from molecules to thoughts--in the common language of coordination dynamics.

Metastability and chimera states in modular delay and pulse-coupled oscillator networks

Modular networks of delay-coupled and pulse-coupled oscillators are presented, which display both transient (metastable) synchronization dynamics and the formation of a large number of "chimera" states characterized by coexistent synchronized and desynchronized subsystems. We consider networks based on both community and small-world topologies. It is shown through simulation that the metastable behaviour of the system is dependent in all cases on connection delay, and a critical region is found that maximizes indices of both metastability and the prevalence of chimera states. We show dependence of phase coherence in synchronous oscillation on the level and strength of external connectivity between communities, and demonstrate that synchronization dynamics are dependent on the modular structure of the network. The long-term behaviour of the system is considered and the relevance of the model briefly discussed with emphasis on biological and neurobiological systems.

Synchronisation effects on the behavioural performance and information dynamics of a simulated minimally cognitive robotic agent

Oscillatory activity is ubiquitous in nervous systems, with solid evidence that synchronisation mechanisms underpin cognitive processes. Nevertheless, its informational content and relationship with behaviour are still to be fully understood. In addition, cognitive systems cannot be properly appreciated without taking into account brain-body- environment interactions. In this paper, we developed a model based on the Kuramoto Model of coupled phase oscillators to explore the role of neural synchronisation in the performance of a simulated robotic agent in two different minimally cognitive tasks. We show that there is a statistically significant difference in performance and evolvability depending on the synchronisation regime of the network. In both tasks, a combination of information flow and dynamical analyses show that networks with a definite, but not too strong, propensity for synchronisation are more able to reconfigure, to organise themselves functionally and to adapt to different behavioural conditions. The results highlight the asymmetry of information flow and its behavioural correspondence. Importantly, it also shows that neural synchronisation dynamics, when suitably flexible and reconfigurable, can generate minimally cognitive embodied behaviour.

Establishing Communication between Neuronal Populations through Competitive Entrainment

The role of gamma frequency oscillation in neuronal interaction, and the relationship between oscillation and information transfer between neurons, has been the focus of much recent research. While the biological mechanisms responsible for gamma oscillation and the properties of resulting networks are well studied, the dynamics of changing phase coherence between oscillating neuronal populations are not well understood. To this end we develop a computational model of competitive selection between multiple stimuli, where the selection and transfer of population-encoded information arises from competition between converging stimuli to entrain a target population of neurons. Oscillation is generated by Pyramidal-Interneuronal Network Gamma through the action of recurrent synaptic connections between a locally connected network of excitatory and inhibitory neurons. Competition between stimuli is driven by differences in coherence of oscillation, while transmission of a single selected stimulus is enabled between generating and receiving neurons via Communication-through-Coherence. We explore the effect of varying synaptic parameters on the competitive transmission of stimuli over different neuron models, and identify a continuous region within the parameter space of the recurrent synaptic loop where inhibition-induced oscillation results in entrainment of target neurons. Within this optimal region we find that competition between stimuli of equal coherence results in model output that alternates between representation of the stimuli, in a manner strongly resembling well-known biological phenomena resulting from competitive stimulus selection such as binocular rivalry.

Chimera states are chaotic transients

Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states," has been described, where in a spatially homogeneous system, regions of irregular incoherent motion coexist with regular synchronized motion, forming a self-organized pattern in a population of nonlocally coupled oscillators. Whereas most previous studies of chimera states focused their attention on the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, here we investigate the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.

Role of local network oscillations in resting-state functional connectivity

Spatio-temporally organized low-frequency fluctuations (<0.1 Hz), observed in BOLD fMRI signal during rest, suggest the existence of underlying network dynamics that emerge spontaneously from intrinsic brain processes. Furthermore, significant correlations between distinct anatomical regions-or functional connectivity (FC)-have led to the identification of several widely distributed resting-state networks (RSNs). This slow dynamics seems to be highly structured by anatomical connectivity but the mechanism behind it and its relationship with neural activity, particularly in the gamma frequency range, remains largely unknown. Indeed, direct measurements of neuronal activity have revealed similar large-scale correlations, particularly in slow power fluctuations of local field potential gamma frequency range oscillations. To address these questions, we investigated neural dynamics in a large-scale model of the human brain's neural activity. A key ingredient of the model was a structural brain network defined by empirically derived long-range brain connectivity together with the corresponding conduction delays. A neural population, assumed to spontaneously oscillate in the gamma frequency range, was placed at each network node. When these oscillatory units are integrated in the network, they behave as weakly coupled oscillators. The time-delayed interaction between nodes is described by the Kuramoto model of phase oscillators, a biologically-based model of coupled oscillatory systems. For a realistic setting of axonal conduction speed, we show that time-delayed network interaction leads to the emergence of slow neural activity fluctuations, whose patterns correlate significantly with the empirically measured FC. The best agreement of the simulated FC with the empirically measured FC is found for a set of parameters where subsets of nodes tend to synchronize although the network is not globally synchronized. Inside such clusters, the simulated BOLD signal between nodes is found to be correlated, instantiating the empirically observed RSNs. Between clusters, patterns of positive and negative correlations are observed, as described in experimental studies. These results are found to be robust with respect to a biologically plausible range of model parameters. In conclusion, our model suggests how resting-state neural activity can originate from the interplay between the local neural dynamics and the large-scale structure of the brain.

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.