Synchrony and clustering in heterogeneous networks with global coupling and parameter dispersion

Recent Progress in Brain Network Models for Medical Applications: A Review

Importance: Pathological perturbations of the brain often spread via connectome to fundamentally alter functional consequences. By integrating multimodal neuroimaging data with mathematical neural mass modeling, brain network models (BNMs) enable to quantitatively characterize aberrant network dynamics underlying multiple neurological and psychiatric disorders. We delved into the advancements of BNM-based medical applications, discussed the prevalent challenges within this field, and provided possible solutions and future directions. Highlights: This paper reviewed the theoretical foundations and current medical applications of computational BNMs. Composed of neural mass models, the BNM framework allows to investigate large-scale brain dynamics behind brain diseases by linking the simulated functional signals to the empirical neurophysiological data, and has shown promise in exploring neuropathological mechanisms, elucidating therapeutic effects, and predicting disease outcome. Despite that several limitations existed, one promising trend of this research field is to precisely guide clinical neuromodulation treatment based on individual BNM simulation. Conclusion: BNM carries the potential to help understand the mechanism underlying how neuropathology affects brain network dynamics, further contributing to decision-making in clinical diagnosis and treatment. Several constraints must be addressed and surmounted to pave the way for its utilization in the clinic.

Diversity-induced trivialization and resilience of neural dynamics

Heterogeneity is omnipresent across all living systems. Diversity enriches the dynamical repertoire of these systems but remains challenging to reconcile with their manifest robustness and dynamical persistence over time, a fundamental feature called resilience. To better understand the mechanism underlying resilience in neural circuits, we considered a nonlinear network model, extracting the relationship between excitability heterogeneity and resilience. To measure resilience, we quantified the number of stationary states of this network, and how they are affected by various control parameters. We analyzed both analytically and numerically gradient and non-gradient systems modeled as non-linear sparse neural networks evolving over long time scales. Our analysis shows that neuronal heterogeneity quenches the number of stationary states while decreasing the susceptibility to bifurcations: a phenomenon known as trivialization. Heterogeneity was found to implement a homeostatic control mechanism enhancing network resilience to changes in network size and connection probability by quenching the system's dynamic volatility.

Interlayer Connectivity Affects the Coherence Resonance and Population Activity Patterns in Two-Layered Networks of Excitatory and Inhibitory Neurons

The firing patterns of neuronal populations often exhibit emergent collective oscillations, which can display substantial regularity even though the dynamics of individual elements is very stochastic. One of the many phenomena that is often studied in this context is coherence resonance, where additional noise leads to improved regularity of spiking activity in neurons. In this work, we investigate how the coherence resonance phenomenon manifests itself in populations of excitatory and inhibitory neurons. In our simulations, we use the coupled FitzHugh-Nagumo oscillators in the excitable regime and in the presence of neuronal noise. Formally, our model is based on the concept of a two-layered network, where one layer contains inhibitory neurons, the other excitatory neurons, and the interlayer connections represent heterotypic interactions. The neuronal activity is simulated in realistic coupling schemes in which neurons within each layer are connected with undirected connections, whereas neurons of different types are connected with directed interlayer connections. In this setting, we investigate how different neurophysiological determinants affect the coherence resonance. Specifically, we focus on the proportion of inhibitory neurons, the proportion of excitatory interlayer axons, and the architecture of interlayer connections between inhibitory and excitatory neurons. Our results reveal that the regularity of simulated neural activity can be increased by a stronger damping of the excitatory layer. This can be accomplished with a higher proportion of inhibitory neurons, a higher fraction of inhibitory interlayer axons, a stronger coupling between inhibitory axons, or by a heterogeneous configuration of interlayer connections. Our approach of modeling multilayered neuronal networks in combination with stochastic dynamics offers a novel perspective on how the neural architecture can affect neural information processing and provide possible applications in designing networks of artificial neural circuits to optimize their function via noise-induced phenomena.

Motif-based mean-field approximation of interacting particles on clustered networks

Interacting particles on graphs are routinely used to study magnetic behavior in physics, disease spread in epidemiology, and opinion dynamics in social sciences. The literature on mean-field approximations of such systems for large graphs typically remains limited to specific dynamics, or assumes cluster-free graphs for which standard approximations based on degrees and pairs are often reasonably accurate. Here, we propose a motif-based mean-field approximation that considers higher-order subgraph structures in large clustered graphs. Numerically, our equations agree with stochastic simulations where existing methods fail.

Optimal responsiveness and information flow in networks of heterogeneous neurons

Cerebral cortex is characterized by a strong neuron-to-neuron heterogeneity, but it is unclear what consequences this may have for cortical computations, while most computational models consider networks of identical units. Here, we study network models of spiking neurons endowed with heterogeneity, that we treat independently for excitatory and inhibitory neurons. We find that heterogeneous networks are generally more responsive, with an optimal responsiveness occurring for levels of heterogeneity found experimentally in different published datasets, for both excitatory and inhibitory neurons. To investigate the underlying mechanisms, we introduce a mean-field model of heterogeneous networks. This mean-field model captures optimal responsiveness and suggests that it is related to the stability of the spontaneous asynchronous state. The mean-field model also predicts that new dynamical states can emerge from heterogeneity, a prediction which is confirmed by network simulations. Finally we show that heterogeneous networks maximise the information flow in large-scale networks, through recurrent connections. We conclude that neuronal heterogeneity confers different responsiveness to neural networks, which should be taken into account to investigate their information processing capabilities.

Dynamics of multiple interacting excitatory and inhibitory populations with delays

A network consisting of excitatory and inhibitory (EI) neurons is a canonical model for understanding local cortical network activity. In this study, we extended the local circuit model and investigated how its dynamical landscape can be enriched when it interacts with another excitatory (E) population with long transmission delays. Through analysis of a rate model and numerical simulations of a corresponding network of spiking neurons, we studied the transition from stationary to oscillatory states by analyzing the Hopf bifurcation structure in terms of two network parameters: (1) transmission delay between the EI subnetwork and the E population and (2) inhibitory couplings that induced oscillatory activity in the EI subnetwork. We found that the critical coupling strength can strongly modulate as a function of transmission delay, and consequently the stationary state can be interwoven intricately with the oscillatory state. Such a dynamical landscape gave rise to an isolated stationary state surrounded by multiple oscillatory states that generated different frequency modes, and cross-frequency coupling developed naturally at the bifurcation points. We identified the network motifs with short- and long-range inhibitory connections that underlie the emergence of oscillatory states with multiple frequencies. Thus, we provided a mechanistic explanation of how the transmission delay to and from the additional E population altered the dynamical landscape. In summary, our results demonstrated the potential role of long-range connections in shaping the network activity of local cortical circuits.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

Heterogeneity of time delays determines synchronization of coupled oscillators

Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.

Relating Alpha Power and Phase to Population Firing and Hemodynamic Activity Using a Thalamo-cortical Neural Mass Model

Oscillations are ubiquitous phenomena in the animal and human brain. Among them, the alpha rhythm in human EEG is one of the most prominent examples. However, its precise mechanisms of generation are still poorly understood. It was mainly this lack of knowledge that motivated a number of simultaneous electroencephalography (EEG) - functional magnetic resonance imaging (fMRI) studies. This approach revealed how oscillatory neuronal signatures such as the alpha rhythm are paralleled by changes of the blood oxygenation level dependent (BOLD) signal. Several such studies revealed a negative correlation between the alpha rhythm and the hemodynamic BOLD signal in visual cortex and a positive correlation in the thalamus. In this study we explore the potential generative mechanisms that lead to those observations. We use a bursting capable Stefanescu-Jirsa 3D (SJ3D) neural-mass model that reproduces a wide repertoire of prominent features of local neuronal-population dynamics. We construct a thalamo-cortical network of coupled SJ3D nodes considering excitatory and inhibitory directed connections. The model suggests that an inverse correlation between cortical multi-unit activity, i.e. the firing of neuronal populations, and narrow band local field potential oscillations in the alpha band underlies the empirically observed negative correlation between alpha-rhythm power and fMRI signal in visual cortex. Furthermore the model suggests that the interplay between tonic and bursting mode in thalamus and cortex is critical for this relation. This demonstrates how biophysically meaningful modelling can generate precise and testable hypotheses about the underpinnings of large-scale neuroimaging signals.

Transitions between dynamical behaviors of oscillator networks induced by diversity of nodes and edges

We study the impact of dynamical and structural heterogeneity on the collective dynamics of large small-world networks of pulse-coupled integrate-and-fire oscillators endowed with refractory periods and time delay. Depending on the choice of homogeneous control parameters (here, refractoriness and coupling strength), these networks exhibit a large spectrum of dynamical behaviors, including asynchronous, partially synchronous, and fully synchronous states. Networks exhibit transitions between these dynamical behaviors upon introducing heterogeneity. We show that the probability for a network to exhibit a certain dynamical behavior (network susceptibility) is affected differently by dynamical and structural heterogeneity and depends on the respective homogeneous dynamics.

Bifurcation and oscillation in a time-delay neural mass model

The neural mass model developed by Lopes da Silva et al. simulates complex dynamics between cortical areas and is able to describe a limit cycle behavior for alpha rhythms in electroencephalography (EEG). In this work, we propose a modified neural mass model that incorporates a time delay. This time-delay model can be used to simulate several different types of EEG activity including alpha wave, interictal EEG, and ictal EEG. We present a detailed description of the model's behavior with bifurcation diagrams. Through simulation and an analysis of the influence of the time delay on the model's oscillatory behavior, we demonstrate that a time delay in neuronal signal transmission could cause seizure-like activity in the brain. Further study of the bifurcations in this new neural mass model could provide a theoretical reference for the understanding of the neurodynamics in epileptic seizures.

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.

Cross-frequency coupling in real and virtual brain networks

Information processing in the brain is thought to rely on the convergence and divergence of oscillatory behaviors of widely distributed brain areas. This information flow is captured in its simplest form via the concepts of synchronization and desynchronization and related metrics. More complex forms of information flow are transient synchronizations and multi-frequency behaviors with metrics related to cross-frequency coupling (CFC). It is supposed that CFC plays a crucial role in the organization of large-scale networks and functional integration across large distances. In this study, we describe different CFC measures and test their applicability in simulated and real electroencephalographic (EEG) data obtained during resting state. For these purposes, we derive generic oscillator equations from full brain network models. We systematically model and simulate the various scenarios of CFC under the influence of noise to obtain biologically realistic oscillator dynamics. We find that (i) specific CFC-measures detect correctly in most cases the nature of CFC under noise conditions, (ii) bispectrum (BIS) and bicoherence (BIC) correctly detect the CFCs in simulated data, (iii) empirical resting state EEG show a prominent delta-alpha CFC as identified by specific CFC measures and the more classic BIS and BIC. This coupling was mostly asymmetric (directed) and generally higher in the eyes closed (EC) than in the eyes open (EO) condition. In conjunction, these two sets of measures provide a powerful toolbox to reveal the nature of couplings from experimental data and as such allow inference on the brain state dependent information processing. Methodological advantages of using CFC measures and theoretical significance of delta and alpha interactions during resting and other brain states are discussed.

Systematic approximations of neural fields through networks of neural masses in the virtual brain

Full brain network models comprise a large-scale connectivity (the connectome) and neural mass models as the network's nodes. Neural mass models absorb implicitly a variety of properties in their constant parameters to achieve a reduction in complexity. In situations, where the local network connectivity undergoes major changes, such as in development or epilepsy, it becomes crucial to model local connectivity explicitly. This leads naturally to a description of neural fields on folded cortical sheets with local and global connectivities. The numerical approximation of neural fields in biologically realistic situations as addressed in Virtual Brain simulations (see http://thevirtualbrain.org/app/ (version 1.0)) is challenging and requires a thorough evaluation if the Virtual Brain approach is to be adapted for systematic studies of disease and disorders. Here we analyze the sampling problem of neural fields for arbitrary dimensions and provide explicit results for one, two and three dimensions relevant to realistically folded cortical surfaces. We characterize (i) the error due to sampling of spatial distribution functions; (ii) useful sampling parameter ranges in the context of encephalographic (EEG, MEG, ECoG and functional MRI) signals; (iii) guidelines for choosing the right spatial distribution function for given anatomical and geometrical constraints.

The Virtual Brain: a simulator of primate brain network dynamics

We present The Virtual Brain (TVB), a neuroinformatics platform for full brain network simulations using biologically realistic connectivity. This simulation environment enables the model-based inference of neurophysiological mechanisms across different brain scales that underlie the generation of macroscopic neuroimaging signals including functional MRI (fMRI), EEG and MEG. Researchers from different backgrounds can benefit from an integrative software platform including a supporting framework for data management (generation, organization, storage, integration and sharing) and a simulation core written in Python. TVB allows the reproduction and evaluation of personalized configurations of the brain by using individual subject data. This personalization facilitates an exploration of the consequences of pathological changes in the system, permitting to investigate potential ways to counteract such unfavorable processes. The architecture of TVB supports interaction with MATLAB packages, for example, the well known Brain Connectivity Toolbox. TVB can be used in a client-server configuration, such that it can be remotely accessed through the Internet thanks to its web-based HTML5, JS, and WebGL graphical user interface. TVB is also accessible as a standalone cross-platform Python library and application, and users can interact with the scientific core through the scripting interface IDLE, enabling easy modeling, development and debugging of the scientific kernel. This second interface makes TVB extensible by combining it with other libraries and modules developed by the Python scientific community. In this article, we describe the theoretical background and foundations that led to the development of TVB, the architecture and features of its major software components as well as potential neuroscience applications.

Inferring network properties of cortical neurons with synaptic coupling and parameter dispersion

Computational models at different space-time scales allow us to understand the fundamental mechanisms that govern neural processes and relate uniquely these processes to neuroscience data. In this work, we propose a novel neurocomputational unit (a mesoscopic model which tell us about the interaction between local cortical nodes in a large scale neural mass model) of bursters that qualitatively captures the complex dynamics exhibited by a full network of parabolic bursting neurons. We observe that the temporal dynamics and fluctuation of mean synaptic action term exhibits a high degree of correlation with the spike/burst activity of our population. With heterogeneity in the applied drive and mean synaptic coupling derived from fast excitatory synapse approximations we observe long term behavior in our population dynamics such as partial oscillations, incoherence, and synchrony. In order to understand the origin of multistability at the population level as a function of mean synaptic coupling and heterogeneity in the firing rate threshold we employ a simple generative model for parabolic bursting recently proposed by Ghosh et al. (2009). Further, we use here a mean coupling formulated for fast spiking neurons for our analysis of generic model. Stability analysis of this mean field network allow us to identify all the relevant network states found in the detailed biophysical model. We derive here analytically several boundary solutions, a result which holds for any number of spikes per burst. These findings illustrate the role of oscillations occurring at slow time scales (bursts) on the global behavior of the network.

Synchrony of two brain regions predicts the blood oxygen level dependent activity of a third

Spontaneously emerging coherent fluctuations have been long observed in electrophysiological and functional magnetic resonance imaging studies. These dynamics have been identified in multiple brain areas in the 1-100 and < 0.1 Hz frequency ranges spanning neurophysiological oscillations and blood oxygen level dependent (BOLD) signals, respectively. In this article, we demonstrate that transient neural synchronization between two sites may lead to the emergence of ultra-slow frequency fluctuations in the BOLD signal at another (third) site. Starting with a network model comprised of three neural oscillators, we illustrate the critical role of time delay and coupling strength in generating these slow coherent fluctuations as a function of intermittently occurring neural coherence. When extending the network toward biologically realistic primate connectivity, we find that the BOLD activation patterns arise from neurophysiological coherence, especially among medial cortical areas. This finding demonstrates a network-level mechanism whereby the BOLD activity at a given region is critically influenced by the neuroelectric synchronization patterns of other regions in the network.

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.

The virtual brain integrates computational modeling and multimodal neuroimaging

Brain function is thought to emerge from the interactions among neuronal populations. Apart from traditional efforts to reproduce brain dynamics from the micro- to macroscopic scales, complementary approaches develop phenomenological models of lower complexity. Such macroscopic models typically generate only a few selected-ideally functionally relevant-aspects of the brain dynamics. Importantly, they often allow an understanding of the underlying mechanisms beyond computational reproduction. Adding detail to these models will widen their ability to reproduce a broader range of dynamic features of the brain. For instance, such models allow for the exploration of consequences of focal and distributed pathological changes in the system, enabling us to identify and develop approaches to counteract those unfavorable processes. Toward this end, The Virtual Brain (TVB) ( www.thevirtualbrain.org ), a neuroinformatics platform with a brain simulator that incorporates a range of neuronal models and dynamics at its core, has been developed. This integrated framework allows the model-based simulation, analysis, and inference of neurophysiological mechanisms over several brain scales that underlie the generation of macroscopic neuroimaging signals. In this article, we describe how TVB works, and we present the first proof of concept.

Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh-Nagumo model

We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of-phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.

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.

Spatiotemporal re-organization of large-scale neural assemblies underlies bimanual coordination

Bimanual coordination engages a distributed network of brain areas, the spatiotemporal organization of which has given rise to intense debates. Do bimanual movements require information processing in the same set of brain areas that are engaged by movements of the individual components (left and right hands)? Or is it necessary that other brain areas are recruited to help in the act of coordination? These two possibilities are often considered as mutually exclusive, with studies yielding support for one or the other depending on techniques and hypotheses. However, as yet there is no account of how the two views may work together dynamically. Using the method of Mode-Level Cognitive Subtraction (MLCS) on high density EEG recorded during unimanual and bimanual movements, we expose spatiotemporal reorganization of large-scale cortical networks during stable inphase and antiphase coordination and transitions between them. During execution of stable bimanual coordination patterns, neural dynamics were dominated by temporal modulation of unimanual networks. At instability and transition, there was evidence for recruitment of additional areas. Our study provides a framework to quantify large-scale network mechanisms underlying complex cognitive tasks often studied with macroscopic neurophysiological recordings.

Ongoing cortical activity at rest: criticality, multistability, and ghost attractors

The ongoing activity of the brain at rest, i.e., under no stimulation and in absence of any task, is astonishingly highly structured into spatiotemporal patterns. These spatiotemporal patterns, called resting state networks, display low-frequency characteristics (<0.1 Hz) observed typically in the BOLD-fMRI signal of human subjects. We aim here to understand the origins of resting state activity through modeling via a global spiking attractor network of the brain. This approach offers a realistic mechanistic model at the level of each single brain area based on spiking neurons and realistic AMPA, NMDA, and GABA synapses. Integrating the biologically realistic diffusion tensor imaging/diffusion spectrum imaging-based neuroanatomical connectivity into the brain model, the resultant emerging resting state functional connectivity of the brain network fits quantitatively best the experimentally observed functional connectivity in humans when the brain network operates at the edge of instability. Under these conditions, the slow fluctuating (<0.1 Hz) resting state networks emerge as structured noise fluctuations around a stable low firing activity equilibrium state in the presence of latent "ghost" multistable attractors. The multistable attractor landscape defines a functionally meaningful dynamic repertoire of the brain network that is inherently present in the neuroanatomical connectivity.

Using large-scale neural models to interpret connectivity measures of cortico-cortical dynamics at millisecond temporal resolution

Over the last two decades numerous functional imaging studies have shown that higher order cognitive functions are crucially dependent on the formation of distributed, large-scale neuronal assemblies (neurocognitive networks), often for very short durations. This has fueled the development of a vast number of functional connectivity measures that attempt to capture the spatiotemporal evolution of neurocognitive networks. Unfortunately, interpreting the neural basis of goal directed behavior using connectivity measures on neuroimaging data are highly dependent on the assumptions underlying the development of the measure, the nature of the task, and the modality of the neuroimaging technique that was used. This paper has two main purposes. The first is to provide an overview of some of the different measures of functional/effective connectivity that deal with high temporal resolution neuroimaging data. We will include some results that come from a recent approach that we have developed to identify the formation and extinction of task-specific, large-scale neuronal assemblies from electrophysiological recordings at a ms-by-ms temporal resolution. The second purpose of this paper is to indicate how to partially validate the interpretations drawn from this (or any other) connectivity technique by using simulated data from large-scale, neurobiologically realistic models. Specifically, we applied our recently developed method to realistic simulations of MEG data during a delayed match-to-sample (DMS) task condition and a passive viewing of stimuli condition using a large-scale neural model of the ventral visual processing pathway. Simulated MEG data using simple head models were generated from sources placed in V1, V4, IT, and prefrontal cortex (PFC) for the passive viewing condition. The results show how closely the conclusions obtained from the functional connectivity method match with what actually occurred at the neuronal network level.

Phase description of spiking neuron networks with global electric and synaptic coupling

Phase models are among the simplest neuron models reproducing spiking behavior, excitability, and bifurcations toward periodic firing. However, coupling among neurons has been considered only using generic arguments valid close to the bifurcation point, and the differentiation between electric and synaptic coupling remains an open question. In this work we aim to address this question and derive a mathematical formulation for the various forms of coupling. We construct a mathematical model based on a planar simplification of the Morris-Lecar model. Based on geometric arguments we then derive a phase description of a network of the above oscillators with biologically realistic electric coupling and subsequently with chemical coupling under fast synapse approximation. We demonstrate analytically that electric and synaptic coupling are differently expressed on the level of the network's phase description with qualitatively different dynamics. Our mathematical analysis shows that a breaking of the translational symmetry in the phase flows is responsible for the different bifurcations paths of electric and synaptic coupling. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network's phase description.

Using the structure of inhibitory networks to unravel mechanisms of spatiotemporal patterning

Neuronal networks exhibit a rich dynamical repertoire, a consequence of both the intrinsic properties of neurons and the structure of the network. It has been hypothesized that inhibitory interneurons corral principal neurons into transiently synchronous ensembles that encode sensory information and subserve behavior. How does the structure of the inhibitory network facilitate such spatiotemporal patterning? We established a relationship between an important structural property of a network, its colorings, and the dynamics it constrains. Using a model of the insect antennal lobe, we show that our description allows the explicit identification of the groups of inhibitory interneurons that switch, during odor stimulation, between activity and quiescence in a coordinated manner determined by features of the network structure. This description optimally matches the perspective of the downstream neurons looking for synchrony in ensembles of presynaptic cells and allows a low-dimensional description of seemingly complex high-dimensional network activity.

Reduced representations of heterogeneous mixed neural networks with synaptic coupling

In the human brain, large-scale neural networks are considered to instantiate the integrative mechanisms underlying higher cognitive, motor, and sensory functions. Computational models of such large-scale networks typically lump thousands of neurons into a functional unit, which serves as the "atom" for the network integration. These atoms display a low dimensional dynamics corresponding to the only type of behavior available for the neurons within the unit, namely, the synchronized regime. Other dynamical features are not part of the unit's repertoire. With this limitation in mind, here we have studied the dynamical behavior of a neural network comprising "all-to-all" synaptically connected excitatory and inhibitory nonidentical neurons. We found that the network exhibits various dynamical characteristics, synchronization being only a particular case. Then we construct a low-dimensional representation of the network dynamics, and we show that this reduced system captures well the main dynamical features of the entire population. Our approach provides an alternate model for a neurocomputational unit of a large-scale network that can account for rich dynamical features of the network at low computational costs.

Neural population modes capture biologically realistic large scale network dynamics

Large scale brain networks are understood nowadays to underlie the emergence of cognitive functions, though the detailed mechanisms are hitherto unknown. The challenges in the study of large scale brain networks are amongst others their high dimensionality requiring significant computational efforts, the complex connectivity across brain areas and the associated transmission delays, as well as the stochastic nature of neuronal processes. To decrease the computational effort, neurons are clustered into neural masses, which then are approximated by reduced descriptions of population dynamics. Here, we implement a neural population mode approach (Assisi et al. in Phys. Rev. Lett. 94(1):018106, 2005; Stefanescu and Jirsa in PLoS Comput. Biol. 4(11):e1000219, 2008), which parsimoniously captures various types of population behavior. We numerically demonstrate that the reduced population mode system favorably captures the high-dimensional dynamics of neuron networks with an architecture involving homogeneous local connectivity and a large-scale, fiber-like connection with time delay.

A method for the estimation of functional brain connectivity from time-series data

A central issue in cognitive neuroscience is which cortical areas are involved in managing information processing in a cognitive task and to understand their temporal interactions. Since the transfer of information in the form of electrical activity from one cortical region will in turn evoke electrical activity in other regions, the analysis of temporal synchronization provides a tool to understand neuronal information processing between cortical regions. We adopt a method for revealing time-dependent functional connectivity. We apply statistical analyses of phases to recover the information flow and the functional connectivity between cortical regions for high temporal resolution data. We further develop an evaluation method for these techniques based on two kinds of model networks. These networks consist of coupled Rössler attractors or of coupled stochastic Ornstein-Uhlenbeck systems. The implemented time-dependent coupling includes uni- and bi-directional connectivities as well as time delayed feedback. The synchronization dynamics of these networks are analyzed using the mean phase coherence, based on averaging over phase-differences, and the general synchronization index. The latter is based on the Shannon entropy. The combination of these with a parametric time delay forms the basis of a connectivity pattern, which includes the temporal and time lagged dynamics of the synchronization between two sources. We model and discuss potential artifacts. We find that the general phase measures are remarkably stable. They produce highly comparable results for stochastic and periodic systems. Moreover, the methods proves useful for identifying brief periods of phase coupling and delays. Therefore, we propose that the method is useful as a basis for generating potential functional connective models.

The membrane response of hippocampal CA3b pyramidal neurons near rest: Heterogeneity of passive properties and the contribution of hyperpolarization-activated currents

Pyramidal neurons in the CA3 region of the hippocampal formation integrate synaptic information arriving in the dendrites within discrete laminar regions. At potentials near or below the resting potential integration of synaptic signals is most affected by the passive properties of the cell and hyperpolarization-activated currents (I(h)). Here we focused specifically on a subset of neurons within the CA3b subregion of the rat hippocampus in order to better understand their membrane response within subthreshold voltage ranges. Using a combined experimental and computational approach we found that the passive properties of these neurons varied up to fivefold between cells. Likewise, there was a large variance in the expression of I(h) channels. However, the contribution of I(h) was minimal at resting potentials endowing the membrane with an apparent linear response to somatic current injection within +/-10 mV. Unlike in CA1 pyramidal neurons, however, I(h) activation was not potentiated in an activity-dependent manner. Computer modeling, based on a combination of voltage- and current-clamp data, suggested that an increasing density of these channels with distance from the soma, compared with a uniform distribution, would have no significant effect on the general properties of the cell because of their relatively lower expression. Nonetheless, temporal summation of excitatory inputs was affected by the presence of I(h) in the dendrites in a frequency- and distance-dependent fashion.

A low dimensional description of globally coupled heterogeneous neural networks of excitatory and inhibitory neurons

Neural networks consisting of globally coupled excitatory and inhibitory nonidentical neurons may exhibit a complex dynamic behavior including synchronization, multiclustered solutions in phase space, and oscillator death. We investigate the conditions under which these behaviors occur in a multidimensional parametric space defined by the connectivity strengths and dispersion of the neuronal membrane excitability. Using mode decomposition techniques, we further derive analytically a low dimensional description of the neural population dynamics and show that the various dynamic behaviors of the entire network can be well reproduced by this reduced system. Examples of networks of FitzHugh-Nagumo and Hindmarsh-Rose neurons are discussed in detail.

Nowadays we know that most cognitive functions are not represented in the brain by the activation of a single area but rather by a complex and rich behavior of brain networks distributed over various cortical and subcortical areas. The communication between brain areas is not instantaneous but also undergoes significant signal transmission delays of up to 100 ms, which increase the computation time for brain network models enormously. In order to allow the efficient investigation of brain network models and their associated cognitive capabilities, we report here a novel, computationally parsimonious, mathematical representation of clusters of neurons. Such reduced clusters are called “neural masses” and serve as nodes in the brain networks. Traditional neural mass descriptions so far allowed only for a very limited repertoire of behaviors, which ultimately rendered their description biologically unrealistic. The neural mass model presented here overcomes this limitation and captures a wide range of dynamic behaviors, but in a computationally efficient reduced form. The integration of novel neural mass models into brain networks represents a step closer toward a computational and biologically realistic realization of brain function.

Noise during rest enables the exploration of the brain's dynamic repertoire

Traditionally brain function is studied through measuring physiological responses in controlled sensory, motor, and cognitive paradigms. However, even at rest, in the absence of overt goal-directed behavior, collections of cortical regions consistently show temporally coherent activity. In humans, these resting state networks have been shown to greatly overlap with functional architectures present during consciously directed activity, which motivates the interpretation of rest activity as day dreaming, free association, stream of consciousness, and inner rehearsal. In monkeys, it has been shown though that similar coherent fluctuations are present during deep anesthesia when there is no consciousness. Here, we show that comparable resting state networks emerge from a stability analysis of the network dynamics using biologically realistic primate brain connectivity, although anatomical information alone does not identify the network. We specifically demonstrate that noise and time delays via propagation along connecting fibres are essential for the emergence of the coherent fluctuations of the default network. The spatiotemporal network dynamics evolves on multiple temporal scales and displays the intermittent neuroelectric oscillations in the fast frequency regimes, 1–100 Hz, commonly observed in electroencephalographic and magnetoencephalographic recordings, as well as the hemodynamic oscillations in the ultraslow regimes, <0.1 Hz, observed in functional magnetic resonance imaging. The combination of anatomical structure and time delays creates a space–time structure in which the neural noise enables the brain to explore various functional configurations representing its dynamic repertoire.

There has been a great deal of interest generated by the observation of resting-state or “default-mode” networks in the human brain. These networks seem to be most engaged when persons are not involved in overt goal-directed behavior. These networks are also thought to underlie certain aspects of conscious introspection and to be specific to humans. Our paper provides a new explanation for rest state fluctuations by suggesting that they reflect a deeper biological principle of organization and are a consequence of the space–time structure of primate anatomical connectivity. In a computational study using a biologically realistic primate cortical connectivity matrix, we show that the rest state networks emerge only if the time delays of signal transmission between brain areas are considered. The combination of anatomical structure and time delays creates a space–time structure in which the neural noise enables the brain to explore various functional configurations representing its dynamic repertoire. The latter repertoire spans temporal scales of multiple orders of magnitude including scales observed in electric potentials and magnetic fields on the scalp, as well as in blood flow signals. Our results provide a testable explanation of the real-world phenomenon of rest state fluctuations in the primate brain.

Mode level cognitive subtraction (MLCS) quantifies spatiotemporal reorganization in large-scale brain topographies

Contemporary brain theories of cognitive function posit spatial, temporal and spatiotemporal reorganization as mechanisms for neural information processing. Corresponding brain imaging results underwrite this perspective of large-scale reorganization. As we show here, a suitable choice of experimental control tasks allows the disambiguation of the spatial and temporal components of reorganization to a quantifiable degree of certainty. When using electro- or magnetoencephalography (EEG or MEG), our approach relies on the identification of lower dimensional spaces obtained from the high dimensional data of suitably chosen control task conditions. Encephalographic data from task conditions are reconstructed within these control spaces. We show that the residual signal (part of the task signal not captured by the control spaces) allows the quantification of the degree of spatial reorganization, such as recruitment of additional brain networks.

Cortical network dynamics with time delays reveals functional connectivity in the resting brain

In absence of all goal-directed behavior, a characteristic network of cortical regions involving prefrontal and cingulate cortices consistently shows temporally coherent fluctuations. The origin of these fluctuations is unknown, but has been hypothesized to be of stochastic nature. In the present paper we test the hypothesis that time delays in the network dynamics play a crucial role in the generation of these fluctuations. By tuning the propagation velocity in a network based on primate connectivity, we scale the time delays and demonstrate the emergence of the resting state networks for biophysically realistic parameters.

Dispersion and time delay effects in synchronized spike-burst networks

We study spike-burst neural activity and investigate its transitions to synchronized states under electrical coupling. Our reported results include the following: (1) Synchronization of spike-burst activity is a multi-time scale phenomenon and burst synchrony is easier to achieve than spike synchrony. (2) Synchrony of networks with time-delayed connections can be achieved at lower coupling strengths than within the same network with instantaneous couplings. (3) The introduction of parameter dispersion into the network destroys the existence of synchrony in the strict sense, but the network dynamics in major regimes of the parameter space can still be effectively captured by a mean field approach if the couplings are excitatory. Our results on synchronization of spiking networks are general of nature and will aid in the development of minimal models of neuronal populations. The latter are the building blocks of large scale brain networks relevant for cognitive processing.

Synchronization and firing death in the dynamics of two interacting excitable units with heterogeneous signals

We study the response of two coupled FitzHugh-Nagumo systems to heterogeneous external inputs. The latter, modeled by periodic parametric stimuli, force the uncoupled excitable systems into a regime of chaotic firing. Due to parameter dispersion involved in randomly distributed amplitudes and/or phases of the external forces the units are nonidentical and their firing events will be asynchronous. Interest is focused on mutually synchronized spikings arising through the coupling. It is demonstrated that the phase difference of the two external forces crucially affects the onset of spike synchronization as well as the resulting degree of synchrony. For large phase differences the degree of spike synchrony is constricted to a maximal possible value and cannot be enhanced upon increasing the coupling strength. We even found that overcritically strong couplings lead to suppression of firing so that the units perform synchronous subthreshold oscillations. This effect, which we call "firing death," is due to a coupling-induced modification of the excitation threshold impeding spiking of the units. In clear contrast, when only the amplitudes of the forces are distributed perfect spike synchrony is achieved for sufficiently strong coupling.

Estimating topology of networks

We suggest a method for estimating the topology of a network based on the dynamical evolution supported on the network. Our method is robust and can be also applied when disturbances and/or modeling errors are presented. Several examples with networks of phase oscillators, pulse-coupled Hindmarch-Rose neurons, and Lorenz oscillators are provided to illustrate our approach.

Consensus in a heterogeneous influence network

We consider a dynamical network model in which a number of agents all move on the plane with the same constant absolute velocity. At each time step, each agent's direction is updated as the average of its direction plus the directions of other agents who can influence it. The influencing capability of each agent is represented by its influencing radius, which is randomly chosen according to a power-law distribution with a scaling exponent between 2 and infinity. As the value of the scaling exponent decreases, the radius distribution becomes more heterogeneous and the network becomes much easier to achieve direction consensus among agents due to the leading roles played by a few hub agents. Furthermore, almost all agents will finally move in the same desired direction in a strong heterogeneous influence network, if and only if a small fraction of hub agents can be controlled to move in the desired direction. These results also reflect the "robust yet fragile" feature of a heterogeneous influence network.

Network reorganization driven by temporal interdependence of its elements

We employ an adaptive parameter control technique based on detection of phase/lag synchrony between the elements of the system to dynamically modify the structure of a network of nonidentical, coupled Rossler oscillators. Two processes are simulated: adaptation, under which the initially different properties of the units converge, and rewiring, in which clusters of interconnected elements are formed based on the temporal correlations. We show how those processes lead to different network structures and investigate their optimal characteristics from the point of view of resulting network properties.

Minimal model for chaotic shear banding in shear thickening fluids

We present a minimal model for spatiotemporal oscillation and rheochaos in shear thickening complex fluids at zero Reynolds number. In the model, a tendency towards inhomogeneous flows in the form of shear bands combines with a slow structural dynamics, modeled by delayed stress relaxation. Using Fourier-space numerics, we study the nonequilibrium "phase diagram" of the fluid as a function of a steady mean (spatially averaged) stress, and of the relaxation time for structural relaxation. We find several distinct regions of periodic behavior (oscillating bands, traveling bands, and more complex oscillations) and also regions of spatiotemporal rheochaos. A low-dimensional truncation of the model retains the important physical features of the full model (including rheochaos) despite the suppression of sharply defined interfaces between shear bands. Our model maps onto the FitzHugh-Nagumo model for neural network dynamics, with an unusual form of long-range coupling.