Synchronization phenomena in human epileptic brain networks

Stability of synchronization manifolds and its nonlinear behaviour in memristive coupled discrete neuron model

In this study, we investigate the impact of first and second-order coupling strengths on the stability of a synchronization manifold in a Discrete FitzHugh-Nagumo (DFHN) neuron model with memristor coupling. Master Stability Function (MSF) is used to estimate the stability of the synchronized manifold. The MSF of the DFHN model exhibits two zero crossings as we vary the coupling strengths, which is categorized as class Γ 2 . Interestingly, both zero-crossing points demonstrate a power-law relationship with respect to both the first-order coupling strength and flux coefficient, as well as the second-order coupling strength and flux coefficient. In contrast, the zero crossings follow a linear relationship between first-order and second-order coupling strength. These linear and nonlinear relationships enable us to forecast the zero-crossing point and, consequently, determine the coupling strengths at which the stability of the synchronization manifold changes for any given set of parameters. We further explore the regime of the stable synchronization manifold within a defined parameter space. Lower values of both first and second-order coupling strengths have minimal impact on the transition between stable and unstable synchronization regimes. Conversely, higher coupling strengths lead to a shrinking regime of the stable synchronization manifold. This reduction follows an exponential relationship with the coupling strengths. This study is helpful in brain-inspired computing systems by understanding synchronization stability in neuron models with memristor coupling. It helps to create more efficient neural networks for tasks like pattern recognition and data processing.

Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves

We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.

Photosensitive Control and Network Synchronization of Chemical Oscillators

The Belousov-Zhabotinsky (BZ) reaction has long been a paradigmatic system for studying chemical oscillations. Here, we experimentally studied the synchronization control within photochemically coupled star networks of BZ oscillators. Experiments were carried out in wells performed in soda-lime glass constructed using novel laser technologies. Utilizing the inherent oscillatory nature of the BZ reaction, we engineered a star network of oscillators interconnected through photochemical inhibitory coupling. Furthermore, the experimental setup presented here could be extrapolated to more complex network architectures with both excitatory and inhibitory couplings, contributing to the fundamental understanding of synchronization in complex systems.

Optimal input reverberation and homeostatic self-organization toward the edge of synchronization

Transient or partial synchronization can be used to do computations, although a fully synchronized network is sometimes related to the onset of epileptic seizures. Here, we propose a homeostatic mechanism that is capable of maintaining a neuronal network at the edge of a synchronization transition, thereby avoiding the harmful consequences of a fully synchronized network. We model neurons by maps since they are dynamically richer than integrate-and-fire models and more computationally efficient than conductance-based approaches. We first describe the synchronization phase transition of a dense network of neurons with different tonic spiking frequencies coupled by gap junctions. We show that at the transition critical point, inputs optimally reverberate through the network activity through transient synchronization. Then, we introduce a local homeostatic dynamic in the synaptic coupling and show that it produces a robust self-organization toward the edge of this phase transition. We discuss the potential biological consequences of this self-organization process, such as its relation to the Brain Criticality hypothesis, its input processing capacity, and how its malfunction could lead to pathological synchronization and the onset of seizure-like activity.

Biomechanical Effects of Seizures on Cerebral Dynamics and Brain Stress

Epilepsy is one of the most common neurological disorders globally, affecting about 50 million people, with nearly 80% of those affected residing in low- and middle-income countries. It is characterized by recurrent seizures that result from abnormal electrical brain activity, with seizures varying widely in manifestation. The exploration of the biomechanical effects that seizures have on brain dynamics and stress levels is relevant for the development of more effective treatments and protective strategies. This study uses a blend of experimental data and computational simulations to assess the brain's physical response during seizures, particularly focusing on the behavior of cerebrospinal fluid and the resulting mechanical stresses on different brain regions. Notable findings show increases in stress, predominantly in the posterior gyri and brainstem, during seizures and an evidence of brain displacement relative to the skull. These observations suggest a dynamic and complex interaction between the brain and skull, with maximum shear stress regions demonstrating the limited yet essential protective role of the CSF. By providing a deeper understanding of the mechanical changes occurring during seizures, this research supports the goal of advancing diagnostic tools, informing more targeted treatment interventions, and guiding the creation of customized therapeutic strategies to enhance neurological care and protect against the adverse effects of seizures.

Self-organized bistability on globally coupled higher-order networks

Self-organized bistability (SOB) stands as a critical behavior for the systems delicately adjusting themselves to the brink of bistability, characterized by a first-order transition. Its essence lies in the inherent ability of the system to undergo enduring shifts between the coexisting states, achieved through the self-regulation of a controlling parameter. Recently, SOB has been established in a scale-free network as a recurrent transition to a short-living state of global synchronization. Here, we embark on a theoretical exploration that extends the boundaries of the SOB concept on a higher-order network (implicitly embedded microscopically within a simplicial complex) while considering the limitations imposed by coupling constraints. By applying Ott-Antonsen dimensionality reduction in the thermodynamic limit to the higher-order network, we derive SOB requirements under coupling limits that are in good agreement with numerical simulations on systems of finite size. We use continuous synchronization diagrams and statistical data from spontaneous synchronized events to demonstrate the crucial role SOB plays in initiating and terminating temporary synchronized events. We show that under weak-coupling consumption, these spontaneous occurrences closely resemble the statistical traits of the epileptic brain functioning.

Synchronization in STDP-driven memristive neural networks with time-varying topology

Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic structural plasticity (HSP) rules remain unclear. Here, we bridge this gap by determining the configurations required to achieve high and stable degrees of complete synchronization (CS) and phase synchronization (PS) in time-varying small-world and random neural networks driven by STDP and HSP. In particular, we found that decreasing P (which enhances the strengthening effect of STDP on the average synaptic weight) and increasing F (which speeds up the swapping rate of synapses between neurons) always lead to higher and more stable degrees of CS and PS in small-world and random networks, provided that the network parameters such as the synaptic time delay [Formula: see text], the average degree [Formula: see text], and the rewiring probability [Formula: see text] have some appropriate values. When [Formula: see text], [Formula: see text], and [Formula: see text] are not fixed at these appropriate values, the degree and stability of CS and PS may increase or decrease when F increases, depending on the network topology. It is also found that the time delay [Formula: see text] can induce intermittent CS and PS whose occurrence is independent F. Our results could have applications in designing neuromorphic circuits for optimal information processing and transmission via synchronization phenomena.

Emergence of chaotic cluster synchronization in heterogeneous networks

Many real-world complex systems rely on cluster synchronization to function properly. A cluster of nodes exhibits synchronous behavior, while others behave erratically. Predicting the emergence of these clusters and understanding the mechanism behind their structure and variation in response to a parameter change is a daunting task in networks that lack symmetry. We unravel the mechanism for the emergence of cluster synchronization in heterogeneous random networks. We develop heterogeneous mean-field approximation together with a self-consistent theory to determine the onset and stability of the cluster. Our analysis shows that cluster synchronization occurs in a wide variety of heterogeneous networks, node dynamics, and coupling functions. The results could lead to a new understanding of the dynamical behavior of networks ranging from neural to social.

Exponentially Long Transient Time to Synchronization of Coupled Chaotic Circle Maps in Dense Random Networks

We study the transition to synchronization in large, dense networks of chaotic circle maps, where an exact solution of the mean-field dynamics in the infinite network and all-to-all coupling limit is known. In dense networks of finite size and link probability of smaller than one, the incoherent state is meta-stable for coupling strengths that are larger than the mean-field critical coupling. We observe chaotic transients with exponentially distributed escape times and study the scaling behavior of the mean time to synchronization.

Sensitive dynamics of brain cognitive networks and its resource constraints

It is well known that brain functions are closely related to the synchronization of brain networks, but the underlying mechanisms are still not completely understood. To study this problem, we here focus on the synchronization of cognitive networks, in contrast to that of a global brain network, as individual brain functions are in fact performed by different cognitive networks but not the global network. In detail, we consider four different levels of brain networks and two approaches, i.e., either with or without resource constraints. For the case of without resource constraints, we find that global brain networks have fundamentally different behaviors from that of the cognitive networks; i.e., the former has a continuous synchronization transition, while the latter shows a novel transition of oscillatory synchronization. This feature of oscillation comes from the sparse links among the communities of cognitive networks, resulting in coupling sensitive dynamics of brain cognitive networks. While for the case of resource constraints, we find that at the global level, the synchronization transition becomes explosive, in contrast to the continuous synchronization for the case of without resource constraints. At the level of cognitive networks, the transition also becomes explosive and the coupling sensitivity is significantly reduced, thus guaranteeing the robustness and fast switch of brain functions. Moreover, a brief theoretical analysis is provided.

Epileptic-network-based prediction and control of seizures in humans

Epilepsy is now conceptualized as a network disease. The epileptic brain network comprises structurally and functionally connected cortical and subcortical brain regions - spanning lobes and hemispheres -, whose connections and dynamics evolve in time. With this concept, focal and generalized seizures as well as other related pathophysiological phenomena are thought to emerge from, spread via, and be terminated by network vertices and edges that also generate and sustain normal, physiological brain dynamics. Research over the last years has advanced concepts and techniques to identify and characterize the evolving epileptic brain network and its constituents on various spatial and temporal scales. Network-based approaches further our understanding of how seizures emerge from the evolving epileptic brain network, and they provide both novel insights into pre-seizure dynamics and important clues for success or failure of measures for network-based seizure control and prevention. In this review, we summarize the current state of knowledge and address several important challenges that would need to be addressed to move network-based prediction and control of seizures closer to clinical translation.

Self-organized bistability on scale-free networks

A dynamical system approaching the first-order transition can exhibit a specific type of critical behavior known as self-organized bistability (SOB). It lies in the fact that the system can permanently switch between the coexisting states under the self-tuning of a control parameter. Many of these systems have a network organization that should be taken into account to understand the underlying processes in detail. In the present paper, we theoretically explore an extension of the SOB concept on the scale-free network under coupling constraints. As provided by the numerical simulations and mean-field approximation in the thermodynamic limit, SOB on scale-free networks originates from facilitated criticality reflected on both macro- and mesoscopic network scales. We establish that the appearance of switches is rooted in spatial self-organization and temporal self-similarity of the network's critical dynamics and replicates extreme properties of epileptic seizure recurrences. Our results, thus, indicate that the proposed conceptual model is suitable to deepen the understanding of emergent collective behavior behind neurological diseases.

Network structure from a characterization of interactions in complex systems

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

Epidemic models characterize seizure propagation and the effects of epilepsy surgery in individualized brain networks based on MEG and invasive EEG recordings

Epilepsy surgery is the treatment of choice for drug-resistant epilepsy patients. However, seizure-freedom is currently achieved in only 2/3 of the patients after surgery. In this study we have developed an individualized computational model based on MEG brain networks to explore seizure propagation and the efficacy of different virtual resections. Eventually, the goal is to obtain individualized models to optimize resection strategy and outcome. We have modelled seizure propagation as an epidemic process using the susceptible-infected (SI) model on individual brain networks derived from presurgical MEG. We included 10 patients who had received epilepsy surgery and for whom the surgery outcome at least one year after surgery was known. The model parameters were tuned in in order to reproduce the patient-specific seizure propagation patterns as recorded with invasive EEG. We defined a personalized search algorithm that combined structural and dynamical information to find resections that maximally decreased seizure propagation for a given resection size. The optimal resection for each patient was defined as the smallest resection leading to at least a 90% reduction in seizure propagation. The individualized model reproduced the basic aspects of seizure propagation for 9 out of 10 patients when using the resection area as the origin of epidemic spreading, and for 10 out of 10 patients with an alternative definition of the seed region. We found that, for 7 patients, the optimal resection was smaller than the resection area, and for 4 patients we also found that a resection smaller than the resection area could lead to a 100% decrease in propagation. Moreover, for two cases these alternative resections included nodes outside the resection area. Epidemic spreading models fitted with patient specific data can capture the fundamental aspects of clinically observed seizure propagation, and can be used to test virtual resections in silico. Combined with optimization algorithms, smaller or alternative resection strategies, that are individually targeted for each patient, can be determined with the ultimate goal to improve surgery outcome. MEG-based networks can provide a good approximation of structural connectivity for computational models of seizure propagation, and facilitate their clinical use.

Functional observability and target state estimation in large-scale networks

Observing the states of a network is fundamental to our ability to explore and control the dynamics of complex natural, social, and technological systems. The problem of determining whether the system is observable has been addressed by network control researchers over the past decade. Progress on the further problem of actually designing and implementing efficient algorithms to infer the states from limited measurements has been hampered by the high dimensionality of large-scale networks. Noting that often only a small number of state variables in a network are essential for control, intervention, and monitoring purposes, this work develops a graph-based theory and highly scalable methods that achieve accurate estimation of target variables of network systems with minimal sensing and computational resources.

The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that a targeted subset of state variables can be reconstructed from the available measurements. Here, we develop a graph-based theory of functional observability, which leads to highly scalable algorithms to 1) determine the minimal set of required sensors and 2) design the corresponding state observer of minimum order. Compared with the full-state observer, the proposed functional observer achieves the same estimation quality with substantially less sensing and fewer computational resources, making it suitable for large-scale networks. We apply the proposed methods to the detection of cyberattacks in power grids from limited phase measurement data and the inference of the prevalence rate of infection during an epidemic under limited testing conditions. The applications demonstrate that the functional observer can significantly scale up our ability to explore otherwise inaccessible dynamical processes on complex networks.

Asynchronous and Coherent Dynamics in Balanced Excitatory-Inhibitory Spiking Networks

Dynamic excitatory-inhibitory (E-I) balance is a paradigmatic mechanism invoked to explain the irregular low firing activity observed in the cortex. However, we will show that the E-I balance can be at the origin of other regimes observable in the brain. The analysis is performed by combining extensive simulations of sparse E-I networks composed of N spiking neurons with analytical investigations of low dimensional neural mass models. The bifurcation diagrams, derived for the neural mass model, allow us to classify the possible asynchronous and coherent behaviors emerging in balanced E-I networks with structural heterogeneity for any finite in-degree K. Analytic mean-field (MF) results show that both supra and sub-threshold balanced asynchronous regimes are observable in our system in the limit N >> K >> 1. Due to the heterogeneity, the asynchronous states are characterized at the microscopic level by the splitting of the neurons in to three groups: silent, fluctuation, and mean driven. These features are consistent with experimental observations reported for heterogeneous neural circuits. The coherent rhythms observed in our system can range from periodic and quasi-periodic collective oscillations (COs) to coherent chaos. These rhythms are characterized by regular or irregular temporal fluctuations joined to spatial coherence somehow similar to coherent fluctuations observed in the cortex over multiple spatial scales. The COs can emerge due to two different mechanisms. A first mechanism analogous to the pyramidal-interneuron gamma (PING), usually invoked for the emergence of γ-oscillations. The second mechanism is intimately related to the presence of current fluctuations, which sustain COs characterized by an essentially simultaneous bursting of the two populations. We observe period-doubling cascades involving the PING-like COs finally leading to the appearance of coherent chaos. Fluctuation driven COs are usually observable in our system as quasi-periodic collective motions characterized by two incommensurate frequencies. However, for sufficiently strong current fluctuations these collective rhythms can lock. This represents a novel mechanism of frequency locking in neural populations promoted by intrinsic fluctuations. COs are observable for any finite in-degree K, however, their existence in the limit N >> K >> 1 appears as uncertain.

Impaired awareness in mesial temporal lobe epilepsy: Network analysis of foramen ovale and scalp EEG

OBJECTIVE

We use co-registration of foramen-ovale and scalp-EEG to investigate network alterations in temporal-lobe epilepsy during focal seizures without (aura) or with impairment of awareness (SIA).

METHODS

One aura and one SIA were selected from six patients. Temporal dynamic among 4 epochs, as well as the differences between aura and SIA, were analyzed through partial directed coherence and graph theory-based indices of centrality.

RESULTS

Regarding the auras temporal evolution, fronto-parietal (FP) regions showed decreased connectivity with respect to the interictal period, in both epileptogenic (EH) and non-epileptogenic hemisphere (nEH). During SIAs, temporal dynamic showed more changes than auras: centrality of mesial temporal (mT) regions changes during all conditions, and nEH FP centrality showed the same dynamic trend of the aura (decreased centrality), until the last epoch, close to the impaired awareness, when showed increased centrality. Comparing SIA with aura, in proximity of impaired awareness, increased centrality was found in all the regions, except in nEH mT.

CONCLUSIONS

Our findings suggested that the impairment of awareness is related to network alterations occurring first in neocortical regions and when awareness is still retained.

SIGNIFICANCE

The analysis of 'hub' alteration can represent a suitable biomarker for scalp EEG-based prediction of awareness impairment.

Synchronization Analysis In Epileptic EEG Signals Via State Transfer Networks Based On Visibility Graph Technique

Epilepsy is a persistent and recurring neurological condition in a community of brain neurons that results from sudden and abnormal electrical discharges. This paper introduces a new form of assessment and interpretation of the changes in electroencephalography (EEG) recordings from different brain regions in epilepsy disorders based on graph analysis and statistical rescale range analysis. In this study, two different states of epilepsy EEG data (preictal and ictal phases), obtained from 17 subjects (18 channels each), were analyzed by a new method called state transfer network (STN). The analysis performed by STN yields a network metric called motifs, which are averaged over all channels and subjects in terms of their persistence level in the network. The results showed an increase of overall motif persistence during the ictal over the preictal phase, reflecting the synchronization increase during the seizure phase (ictal). An evaluation of intermotif cross-correlation indicated a definite manifestation of such synchronization. Moreover, these findings are compared with several other well-known methods such as synchronization likelihood (SL), visibility graph similarity (VGS), and global field synchronization (GFS). It is hinted that the STN method is in good agreement with approaches in the literature and more efficient. The most significant contribution of this research is introducing a novel nonlinear analysis technique of generalized synchronization. The STN method can be used for classifying epileptic seizures based on the synchronization changes between multichannel data.

Time in Brain: How Biological Rhythms Impact on EEG Signals and on EEG-Derived Brain Networks

Electroencephalography (EEG) is a widely employed tool for exploring brain dynamics and is used extensively in various domains, ranging from clinical diagnosis via neuroscience, cognitive science, cognitive psychology, psychophysiology, neuromarketing, neurolinguistics, and pharmacology to research on brain computer interfaces. EEG is the only technique that enables the continuous recording of brain dynamics over periods of time that range from a few seconds to hours and days and beyond. When taking long-term recordings, various endogenous and exogenous biological rhythms may impinge on characteristics of EEG signals. While the impact of the circadian rhythm and of ultradian rhythms on spectral characteristics of EEG signals has been investigated for more than half a century, only little is known on how biological rhythms influence characteristics of brain dynamics assessed with modern EEG analysis techniques. At the example of multiday, multichannel non-invasive and invasive EEG recordings, we here discuss the impact of biological rhythms on temporal changes of various characteristics of human brain dynamics: higher-order statistical moments and interaction properties of multichannel EEG signals as well as local and global characteristics of EEG-derived evolving functional brain networks. Our findings emphasize the need to take into account the impact of biological rhythms in order to avoid erroneous statements about brain dynamics and about evolving functional brain networks.

Asymmetry-induced isolated fully synchronized state in coupled oscillator populations

A symmetry-breaking mechanism is investigated that creates bistability between fully and partially synchronized states in oscillator networks. Two populations of oscillators with unimodal frequency distribution and different amplitudes, in the presence of weak global coupling, are shown to simplify to a modular network with asymmetrical coupling. With increasing the coupling strength, a synchronization transition is observed with an isolated fully synchronized state. The results are interpreted theoretically in the thermodynamic limit and confirmed in experiments with chemical oscillators.

Different Modes of Low-Frequency Focused Ultrasound-Mediated Attenuation of Epilepsy Based on the Topological Theory

Epilepsy is common brain dysfunction, where abnormal synchronized activities can be observed across multiple brain regions. Low-frequency focused pulsed ultrasound has been proven to modulate the epileptic brain network. In this study, we used two modes of low-intensity focused ultrasound (pulsed-wave and continuous-wave) to sonicate the brains of KA-induced epileptic rats, analyzed the EEG functional brain connections to explore their respective effect on the epileptic brain network, and discuss the mechanism of ultrasound neuromodulation. By comparing the brain network characteristics before and after sonication, we found that two modes of ultrasound both significantly affected the functional brain network, especially in the low-frequency band below 12 Hz. After two modes of sonication, the power spectral density of the EEG signals and the connection strength of the brain network were significantly reduced, but there was no significant difference between the two modes. Our results indicated that the ultrasound neuromodulation could effectively regulate the epileptic brain connections. The ultrasound-mediated attenuation of epilepsy was independent of modes of ultrasound.

Suppression of chaotic bursting synchronization in clustered scale-free networks by an external feedback signal

Oscillatory activities in the brain, detected by electroencephalograms, have identified synchronization patterns. These synchronized activities in neurons are related to cognitive processes. Additionally, experimental research studies on neuronal rhythms have shown synchronous oscillations in brain disorders. Mathematical modeling of networks has been used to mimic these neuronal synchronizations. Actually, networks with scale-free properties were identified in some regions of the cortex. In this work, to investigate these brain synchronizations, we focus on neuronal synchronization in a network with coupled scale-free networks. The networks are connected according to a topological organization in the structural cortical regions of the human brain. The neuronal dynamic is given by the Rulkov model, which is a two-dimensional iterated map. The Rulkov neuron can generate quiescence, tonic spiking, and bursting. Depending on the parameters, we identify synchronous behavior among the neurons in the clustered networks. In this work, we aim to suppress the neuronal burst synchronization by the application of an external perturbation as a function of the mean-field of membrane potential. We found that the method we used to suppress synchronization presents better results when compared to the time-delayed feedback method when applied to the same model of the neuronal network.

Extreme synchronization events in a Kuramoto model: The interplay between resource constraints and explosive transitions

Many living and artificial systems possess structural and dynamical properties of complex networks. One of the most exciting living networked systems is the brain, in which synchronization is an essential mechanism of its normal functioning. On the other hand, excessive synchronization in neural networks reflects undesired pathological activity, including various forms of epilepsy. In this context, network-theoretical approach and dynamical modeling may uncover deep insight into the origins of synchronization-related brain disorders. However, many models do not account for the resource consumption needed for the neural networks to synchronize. To fill this gap, we introduce a phenomenological Kuramoto model evolving under the excitability resource constraints. We demonstrate that the interplay between increased excitability and explosive synchronization induced by the hierarchical organization of the network forces the system to generate short-living extreme synchronization events, which are well-known signs of epileptic brain activity. Finally, we establish that the network units occupying the medium levels of hierarchy most strongly contribute to the birth of extreme events emphasizing the focal nature of their origin.

Graph energy based centrality measures to detect epileptogenic focal invasive EEG electrodes

PURPOSE

Medically intractable epilepsy can be treated with surgical interventions, which require localization of the cortical region where seizures start. This region is referred to as the epileptogenic zone (EZ). Good surgical outcomes depend on an exact localization of the EZ.

METHODS

We propose a graph theoretical approach providing a novel method to localize the epileptogenic zone using invasive electroencephalogram (EEG) data. The proposed methods employ centrality determination using three graph energies, namely simple graph energy, Laplacian energy, and distance energy. Centrality values of invasive EEG electrodes from 19 patients were analyzed at different frequency bands and at different time points. K-means clustering was used to distinguish focal (electrodes placed in the epileptogenic zone) from non-focal electrodes using the centrality values obtained.

RESULTS

Focal electrodes show higher centrality values when compared to non-focal electrodes. All three graph energy based centrality measures proposed show maximum f-score and accuracy during the early seizure phase in the gamma frequency band. Among the three proposed methods, simple graph energy based centrality outperforms Laplacian centrality and distance energy based centrality and also other related and competitive methods available in the literature in terms of accuracy and f-score.

CONCLUSION

Graph energy based centrality measures are useful parameters for the delineation of the epileptogenic zone. Among the three centrality measures examined, simple graph energy based centrality proved best suited for this purpose.

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

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

Validation of virtual resection on intraoperative interictal data acquired during epilepsy surgery

OBJECTIVE

A 'Virtual resection' consists of computationally simulating the effect of an actual resection on the brain. We validated two functional connectivity based virtual resection methods with the actual connectivity measured using post-resection intraoperative recordings.

METHODS

A non-linear association index was applied to pre-resection recordings from 11 extra-temporal focal epilepsy patients. We computed two virtual resection strategies: first, a 'naive' one obtained by simply removing from the connectivity matrix the electrodes that were resected; second, a virtual resection with partialization accounting for the influence of resected electrodes on not-resected electrodes. We validated the virtual resections with two analysis: 1) We tested with a Kolmogorov-Smirnov test if the distributions of connectivity values after the virtual resections differed from the actual post-resection connectivity distribution; 2) we tested if the overall effect of the resection measured by contrasting pre-resection and post-resection connectivity values is detectable with the virtual resection approach using a Kolmogorv-Smirnov test.

RESULTS

The estimation of post-resection connectivity values did not succeed for both methods. In the second analysis, the naive method failed completely to detect the effect found between pre-resection and post-resection connectivity distributions, while the partialization method agreed with post-resection measurements in detecting a drop connectivity compared to pre-resection recordings.

CONCLUSION

Our findings suggest that the partialization technique is superior to the naive method in detecting the overall effect after the resection.

SIGNIFICANCE

We pointed out how a realistic validation based on actual post-resection recordings reveals that virtual resection methods are not yet mature to inform the clinical decision-making.

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

The role of node dynamics in shaping emergent functional connectivity patterns in the brain

The contribution of structural connectivity to functional brain states remains poorly understood. We present a mathematical and computational study suited to assess the structure–function issue, treating a system of Jansen–Rit neural mass nodes with heterogeneous structural connections estimated from diffusion MRI data provided by the Human Connectome Project. Via direct simulations we determine the similarity of functional (inferred from correlated activity between nodes) and structural connectivity matrices under variation of the parameters controlling single-node dynamics, highlighting a nontrivial structure–function relationship in regimes that support limit cycle oscillations. To determine their relationship, we firstly calculate network instabilities giving rise to oscillations, and the so-called ‘false bifurcations’ (for which a significant qualitative change in the orbit is observed, without a change of stability) occurring beyond this onset. We highlight that functional connectivity (FC) is inherited robustly from structure when node dynamics are poised near a Hopf bifurcation, whilst near false bifurcations, and structure only weakly influences FC. Secondly, we develop a weakly coupled oscillator description to analyse oscillatory phase-locked states and, furthermore, show how the modular structure of FC matrices can be predicted via linear stability analysis. This study thereby emphasises the substantial role that local dynamics can have in shaping large-scale functional brain states.

Patterns of oscillation across the brain arise because of structural connections between brain regions. However, the type of oscillation at a site may also play a contributory role. We focus on an idealised model of a neural mass network, coupled using estimates of structural connections obtained via tractography on Human Connectome Project MRI data. Using a mixture of computational and mathematical techniques, we show that functional connectivity is inherited most strongly from structural connectivity when the network nodes are poised at a Hopf bifurcation. However, beyond the onset of this oscillatory instability a phase-locked network state can undergo a false bifurcation, and structural connectivity only weakly influences functional connectivity. This highlights the important effect that local dynamics can have on large-scale brain states.

Analysis and data-driven reconstruction of bivariate jump-diffusion processes

We introduce the bivariate jump-diffusion process, consisting of two-dimensional diffusion and two-dimensional jumps, that can be coupled to one another. We present a data-driven, nonparametric estimation procedure of higher-order (up to 8) Kramers-Moyal coefficients that allows one to reconstruct relevant aspects of the underlying jump-diffusion processes and to recover the underlying parameters. The procedure is validated with numerically integrated data using synthetic bivariate time series from continuous and discontinuous processes. We further evaluate the possibility of estimating the parameters of the jump-diffusion model via data-driven analyses of the higher-order Kramers-Moyal coefficients, and the limitations arising from the scarcity of points in the data or disproportionate parameters in the system.

Evaluation of Granger Causality Measures for Constructing Networks from Multivariate Time Series

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. These measures are in the time domain, such as model-based and information measures, the frequency domain, and the phase domain. The study aims also to compare bivariate and multivariate measures, linear and nonlinear measures, as well as the use of dimension reduction in linear model-based measures and information measures. The latter is particular relevant in the study of high-dimensional time series. For the performance of the multivariate causality measures, low and high dimensional coupled dynamical systems are considered in discrete and continuous time, as well as deterministic and stochastic. The measures are evaluated and ranked according to their ability to provide causality networks that match the original coupling structure. The simulation study concludes that the Granger causality measures using dimension reduction are superior and should be preferred particularly in studies involving many observed variables, such as multi-channel electroencephalograms and financial markets.

Causal stability and synchronization

Synchronization of chaos arises between coupled dynamical systems and is very well understood as a temporal phenomenon, which leads the coupled systems to converge or develop a dependence with time. In this work, we provide a complementary spatial perspective to this phenomenon by introducing the novel idea of causal stability. We then propose and prove a causal stability synchronization theorem as a necessary and sufficient condition for complete synchronization. We also provide an empirical criterion to identify synchronizing variables in coupled identical chaotic dynamical systems based on intrasystem causal influences estimated using time series data of the driving system alone. For this, a recently proposed measure, Compression-Complexity Causality (CCC), is used. The sign and magnitude of the estimated CCC value capture the nature of dynamical influences from each variable to rest of the subsystem and are thus able to determine whether or not the variable, when used to couple another system, will drive that system to synchronization.

Role of inhibitory control in modulating focal seizure spread

Focal seizure propagation is classically thought to be spatially contiguous. However, distribution of seizures through a large-scale epileptic network has been theorized. Here, we used a multielectrode array, wide field calcium imaging, and two-photon calcium imaging to study focal seizure propagation pathways in an acute rodent neocortical 4-aminopyridine model. Although ictal neuronal bursts did not propagate beyond a 2-3-mm region, they were associated with hemisphere-wide field potential fluctuations and parvalbumin-positive interneuron activity outside the seizure focus. While bicuculline surface application enhanced contiguous seizure propagation, focal bicuculline microinjection at sites distant to the 4-aminopyridine focus resulted in epileptic network formation with maximal activity at the two foci. Our study suggests that both classical and epileptic network propagation can arise from localized inhibition defects, and that the network appearance can arise in the context of normal brain structure without requirement for pathological connectivity changes between sites.

Traceability and dynamical resistance of precursor of extreme events

Extreme events occur in a variety of natural, technical, and societal systems and often have catastrophic consequences. Their low-probability, high-impact nature has recently triggered research into improving our understanding of generating mechanisms, providing early warnings as well as developing control strategies. For the latter to be effective, knowledge about dynamical resistance of a system prior to an extreme event is of utmost importance. Here we introduce a novel time-series-based and non-perturbative approach to efficiently monitor dynamical resistance and apply it to high-resolution observations of brain activities from 43 subjects with uncontrollable epileptic seizures. We gain surprising insights into pre-seizure dynamical resistance of brains that also provide important clues for success or failure of measures for seizure prevention. The novel resistance monitoring perspective advances our understanding of precursor dynamics in complex spatio-temporal systems with potential applications in refining control strategies.

Recurrence quantification analysis for the identification of burst phase synchronisation

In this work, we apply the spatial recurrence quantification analysis (RQA) to identify chaotic burst phase synchronisation in networks. We consider one neural network with small-world topology and another one composed of small-world subnetworks. The neuron dynamics is described by the Rulkov map, which is a two-dimensional map that has been used to model chaotic bursting neurons. We show that with the use of spatial RQA, it is possible to identify groups of synchronised neurons and determine their size. For the single network, we obtain an analytical expression for the spatial recurrence rate using a Gaussian approximation. In clustered networks, the spatial RQA allows the identification of phase synchronisation among neurons within and between the subnetworks. Our results imply that RQA can serve as a useful tool for studying phase synchronisation even in networks of networks.

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

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

MAIN RESULTS

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

SIGNIFICANCE

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

Detecting Abnormal Pattern of Epileptic Seizures via Temporal Synchronization of EEG Signals

OBJECTIVE

Synchronization phenomena of epileptic electroencephalography (EEG) have long been studied. In this study, we aim at investigating the spatial-temporal synchronization pattern in epileptic human brains using the spectral graph theoretic features extracted from scalp EEG and developing an efficient multivariate approach for detecting seizure onsets in real time.

METHODS

A complex network model is used for representing the recurrence pattern of EEG signals, based on which the temporal synchronization patterns are quantified using the spectral graph theoretic features. Furthermore, a statistical control chart is applied to the extracted features overtime for monitoring the transits from normal to epileptic states in multivariate EEG systems.

RESULTS

Our method is tested on 23 patients from CHB-MIT Scalp EEG database. The results show that the graph theoretic feature yields a high sensitivity (  ∼ 98%) and low latency (  ∼ 6 s) on average, and seizure onsets in 18 patients are 100% detected.

CONCLUSION

Our approach validates the increased temporal synchronization in epileptic EEG and achieves a comparable detection performance to previous studies.

SIGNIFICANCE

We characterize the temporal synchronization patterns of epileptic EEG using spectral network metrics. In addition, we found significant changes in temporal synchronization in epileptic EEG, which enable a patient-specific approach for real-time seizure detection for personalized diagnosis and treatment.

Evaluating resective surgery targets in epilepsy patients: A comparison of quantitative EEG methods

BACKGROUND

Quantitative analysis of intracranial EEG is a promising tool to assist clinicians in the planning of resective brain surgery in patients suffering from pharmacoresistant epilepsies. Quantifying the accuracy of such tools, however, is nontrivial as a ground truth to verify predictions about hypothetical resections is missing.

NEW METHOD

As one possibility to address this, we use customized hypotheses tests to examine the agreement of the methods on a common set of patients. One method uses machine learning techniques to enable the predictive modeling of EEG time series. The other estimates nonlinear interrelation between EEG channels. Both methods were independently shown to distinguish patients with excellent post-surgical outcome (Engel class I) from those without improvement (Engel class IV) when assessing the electrodes associated with the tissue that was actually resected during brain surgery. Using the AND and OR conjunction of both methods we evaluate the performance gain that can be expected when combining them.

RESULTS

Both methods' assessments correlate strongly positively with the similarity between a hypothetical resection and the corresponding actual resection in class I patients. Moreover, the Spearman rank correlation between the methods' patient rankings is significantly positive.

COMPARISON WITH EXISTING METHOD(S)

To our best knowledge, this is the first study comparing surgery target assessments from fundamentally differing techniques.

CONCLUSIONS

Although conceptually completely independent, there is a relation between the predictions obtained from both methods. Their broad consensus supports their application in clinical practice to provide physicians additional information in the process of presurgical evaluation.

Critical and Supercritical Spatiotemporal Calcium Dynamics in Beta Cells

A coordinated functioning of beta cells within pancreatic islets is mediated by oscillatory membrane depolarization and subsequent changes in cytoplasmic calcium concentration. While gap junctions allow for intraislet information exchange, beta cells within islets form complex syncytia that are intrinsically nonlinear and highly heterogeneous. To study spatiotemporal calcium dynamics within these syncytia, we make use of computational modeling and confocal high-speed functional multicellular imaging. We show that model predictions are in good agreement with experimental data, especially if a high degree of heterogeneity in the intercellular coupling term is assumed. In particular, during the first few minutes after stimulation, the probability distribution of calcium wave sizes is characterized by a power law, thus indicating critical behavior. After this period, the dynamics changes qualitatively such that the number of global intercellular calcium events increases to the point where the behavior becomes supercritical. To better mimic normal in vivo conditions, we compare the described behavior during supraphysiological non-oscillatory stimulation with the behavior during exposure to a slightly lower and oscillatory glucose challenge. In the case of this protocol, we observe only critical behavior in both experiment and model. Our results indicate that the loss of oscillatory changes, along with the rise in plasma glucose observed in diabetes, could be associated with a switch to supercritical calcium dynamics and loss of beta cell functionality.

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

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

Reliability of EEG Interactions Differs between Measures and Is Specific for Neurological Diseases

Alterations of interaction (connectivity) of the EEG reflect pathological processes in patients with neurologic disorders. Nevertheless, it is questionable whether these patterns are reliable over time in different measures of interaction and whether this reliability of the measures is the same across different patient populations. In order to address this topic we examined 22 patients with mild cognitive impairment, five patients with subjective cognitive complaints, six patients with right-lateralized temporal lobe epilepsy, seven patients with left lateralized temporal lobe epilepsy, and 20 healthy controls. We calculated 14 measures of interaction from two EEG-recordings separated by 2 weeks. In order to characterize test-retest reliability, we correlated these measures for each group and compared the correlations between measures and between groups. We found that both measures of interaction as well as groups differed from each other in terms of reliability. The strongest correlation coefficients were found for spectrum, coherence, and full frequency directed transfer function (average rho > 0.9). In the delta (2–4 Hz) range, reliability was lower for mild cognitive impairment compared to healthy controls and left lateralized temporal lobe epilepsy. In the beta (13–30 Hz), gamma (31–80 Hz), and high gamma (81–125 Hz) frequency ranges we found decreased reliability in subjective cognitive complaints compared to mild cognitive impairment. In the gamma and high gamma range we found increased reliability in left lateralized temporal lobe epilepsy patients compared to healthy controls. Our results emphasize the importance of documenting reliability of measures of interaction, which may vary considerably between measures, but also between patient populations. We suggest that studies claiming clinical usefulness of measures of interaction should provide information on the reliability of the results. In addition, differences between patient groups in reliability of interactions in the EEG indicate the potential of reliability to serve as a new biomarker for pathological memory decline as well as for epilepsy. While the brain concert of information flow is generally variable, high reliability, and thus, low variability may reflect abnormal firing patterns.

The role of driver nodes in managing epileptic seizures: Application of Kuramoto model

Synchronization is an important global phenomenon which could be found in a wide range of complex systems such as brain or electronic devices. However, in some circumstances the synchronized states are not desirable for the system and should be suppressed. For example, excessively synchronized activities in the brain network could be the root of neuronal disorders like epileptic seizures. According to the controllability theory of the complex networks, a minimum set of driver nodes has the ability to control the entire system. In this study, we examine the role of driver nodes in suppressing the excessive synchronization in a generalized Kuramoto model, which consists of two types of oscillators: contrarian and regular ones. We used two different structural topologies: Barabási-Albert scale-free (BASF) network and Caenorhabditis elegans (C.elegans) neuronal network. Our results show that contrarian driver nodes have the sufficient ability to break the synchronized level of the systems. In this case, the system coherency level is not fully suppressed that is avoiding dysfunctions of normal brain functions which require the neuronal synchronized activities. Moreover, in this case, the oscillators grouped in two distinct synchronized clusters that could be an indication of chaotic behavior of the system known as resting-state activity of the brain.