Improved time complexity for spintronic oscillator Ising machines compared to a popular classical optimization algorithm for the Max-Cut problem.

Solving certain combinatorial optimization problems like Max-Cut becomes challenging once the graph size and edge connectivity increase beyond a threshold, with brute-force algorithms which solve such problems exactly on conventional digital computers having the bottleneck of exponential time complexity. Hence currently, such problems are instead solved approximately using algorithms like Goemans-Williamson (GW) algorithm, run on conventional computers with polynomial time complexity. Phase binarized oscillators (PBOs), also often known as oscillator Ising machines (OIM), have been proposed as an alternative to solve such problems. In this paper, restricting ourselves to the combinatorial optimization problem Max-Cut solved on three kinds of graphs (Mobius Ladder, random cubic, Erdos Renyi) up to 100 nodes, we empirically show that computation time/ time to solution (TTS) for PBOs (captured through Kuramoto model) grows at a much lower rate (logarithmically: O(log(N)), with respect to graph size N) compared to GW algorithm, for which TTS increases as square of graph size (O(N2). However, Kuramoto model being a physics-agnostic mathematical model, this time complexity/ TTS trend for PBOs is a general trend and is device-physics agnostic. So for more specific results, we choose spintronic oscillators, known for their high operating frequency (in GHz), and model them through Slavin's model which captures the physics of their coupled magnetization oscillation dynamics. We thereby empirically show that TTS of spintronic oscillators also grows logarithmically with graph size (O(log(N)), while their accuracy is comparable to that of GW. So spintronic oscillators have improved time complexity over GW algorithm. For large graphs, they are expected to compute Max-Cut values much faster than GW algorithm, as well as other oscillators operating at lower frequencies, while maintaining the same level of accuracy. .

A modification to the Kuramoto model to simulate epileptic seizures as synchronization.

The Kuramoto model was developed to describe the coupling of oscillators, motivated by the natural synchronization phenomena. We are interested in modeling an epileptic seizure considering it as the synchronization of action potentials using and modifying this model. In this article, we propose to modify this model, changing the constant coupling force for a function with logistic growth to simulate the onset and epileptic seizure level in an adult male rat caused by the administration of lithium-pilocarpine. Later, we select some frequencies and their respective amplitude values using an algorithm based on the fast Fourier transform (FFT) from an electroencephalography signal when the rat is in basal conditions. Then, we take these values as the natural frequencies of the oscillators in the modified Kuramoto model, considering every oscillator as a single neuron to simulate the emergence of an epileptic seizure numerically by increasing the synchronization value in the coupling function. Finally, using Dynamic Time Warping algorithm, we compare the simulated signal by the Kuramoto model with an FFT approximation of the epileptic seizure.

False ST-segment elevation by artifact due to an infusion pump.

A patient admitted for non-ST-elevation acute coronary syndrome showed an episode of ST-segment elevation on the monitor. These alterations were due to an artifact produced by the administration of a saline bolus through an infusion pump that disappeared at the end of the bolus. Our findings highlight that the interpretation of the electrocardiogram requires careful analysis and correlation with the clinical situation and with other physiological parameters.

An interdisciplinary computational model for predicting traumatic brain injury: Linking biomechanics and functional neural networks.

The brain is a complex network consisting of neuron cell bodies in the gray matter and their axonal projections, forming the white matter tracts. These neurons are supported by an equally complex vascular network as well as glial cells. Traumatic brain injury (TBI) can lead to the disruption of the structural and functional brain networks due to disruption of both neuronal cell bodies in the gray matter as well as their projections and supporting cells. To explore how an impact can alter the function of brain networks, we integrated a finite element (FE) brain mechanics model with linked models of brain dynamics (Kuramoto oscillator) and vascular perfusion (Balloon-Windkessel) in this study. We used empirical resting-state functional magnetic resonance imaging (MRI) data to optimize the fit of our brain dynamics and perfusion models to clinical data. Results from the FE model were used to mimic injury in these optimized brain dynamics models: injury to the nodes (gray matter) led to a decrease in the nodal oscillation frequency, while damage to the edges (axonal connections/white matter) progressively decreased coupling among connected nodes. A total of 53 cases, including 33 non-injurious and 20 concussive head impacts experienced by professional American football players were simulated using this integrated model. We examined the correlation of injury outcomes with global measures of structural connectivity, neural dynamics, and functional connectivity of the brain networks when using different lesion methods. Results show that injurious head impacts cause significant alterations in global network topology regardless of lesion methods. Changes between the disrupted and healthy functional connectivity (measured by Pearson correlation) consistently correlated well with injury outcomes (AUC≥0.75), although the predictive performance is not significantly different (p>0.05) to that of traditional kinematic measures (angular acceleration). Intriguingly, our lesion model for gray matter damage predicted increases in global efficiency and clustering coefficient with increases in injury risk, while disrupting axonal connections led to lower network efficiency and clustering. When both injury mechanisms were combined into a single injury prediction model, the injury prediction performance depended on the thresholds used to determine neurodegeneration and mechanical tolerance for axonal injury. Together, these results point towards complex effects of mechanical trauma to the brain and provide a new framework for understanding brain injury at a causal mechanistic level and developing more effective diagnostic methods and therapeutic interventions.

A neurodynamic model of inter-brain coupling in the gamma band.

The use of EEG to simultaneously record multiple brains (i.e., hyperscanning) during social interactions has led to the discovery of inter-brain coupling (IBC). IBC is defined as the neural synchronization between people and is considered to be a marker of social interaction. IBC has previously been observed across different frequency bands, including theta [4-7 Hz]. Given the proximity of this frequency range with behavioral rhythms, models have been able to combine IBC in theta with sensorimotor coordination patterns. Interestingly, empirical EEG-hyperscanning results also report the emergence of IBC in the gamma range [>30 Hz]. Gamma oscillations' fast and transient nature makes a direct link between gamma-IBC and other (much slower) interpersonal dynamics difficult, leaving gamma-IBC without a plausible model. However, at the intrabrain level, gamma activity is coupled with the dynamics of lower frequencies through cross-frequency coupling (CFC). This paper provides a biophysical explanation, through the simulation of neural data, for the emergence of gamma inter-brain coupling using a Kuramoto model of four oscillators divided into two separate (brain) units. By modulating both the degree of inter-brain coupling in the theta band (i.e., between-units coupling) and CFC (i.e., intraunit theta-gamma coupling), we provide a theoretical explanation of the observed gamma-IBC phenomenon in the EEG-hyperscanning literature.NEW & NOTEWORTHY The last years were marked by an increasing interest in multiple-brain recordings. However, the inter-brain coupling arising across interacting individuals also sparks debates about the underlying biological mechanisms. The inter-brain coupling in the gamma band [>30 Hz] was particularly criticized for lacking a theoretical framework. Here, by using biologically informed neural simulations with the Kuramoto model, we assess the role of intra- and inter-brain neural dynamics in the emergence of inter-brain synchrony in the gamma band.

Tractography density affects whole-brain structural architecture and resting-state dynamical modeling.

Dynamical modeling of the resting-state brain dynamics essentially relies on the empirical neuroimaging data utilized for the model derivation and validation. There is however still no standardized data processing for magnetic resonance imaging pipelines and the structural and functional connectomes involved in the models. In this study, we thus address how the parameters of diffusion-weighted data processing for structural connectivity (SC) can influence the validation results of the whole-brain mathematical models informed by SC. For this, we introduce a set of simulation conditions including the varying number of total streamlines of the whole-brain tractography (WBT) used for extraction of SC, cortical parcellations based on functional and anatomical brain properties and distinct model fitting modalities. The main objective of this study is to explore how the quality of the model validation can vary across the considered simulation conditions. We observed that the graph-theoretical network properties of structural connectome can be affected by varying tractography density and strongly relate to the model performance. We also found that the optimal number of the total streamlines of WBT can vary for different brain atlases. Consequently, we suggest a way how to improve the model performance based on the network properties and the optimal parameter configurations from multiple WBT conditions. Furthermore, the population of subjects can be stratified into subgroups with divergent behaviors induced by the varying WBT density such that different recommendations can be made with respect to the data processing for individual subjects and brain parcellations.

Stochastic synchronization of dynamics on the human connectome.

Synchronization is a collective mechanism by which oscillatory networks achieve their functions. Factors driving synchronization include the network's topological and dynamical properties. However, how these factors drive the emergence of synchronization in the presence of potentially disruptive external inputs like stochastic perturbations is not well understood, particularly for real-world systems such as the human brain. Here, we aim to systematically address this problem using a large-scale model of the human brain network (i.e., the human connectome). The results show that the model can produce complex synchronization patterns transitioning between incoherent and coherent states. When nodes in the network are coupled at some critical strength, a counterintuitive phenomenon emerges where the addition of noise increases the synchronization of global and local dynamics, with structural hub nodes benefiting the most. This stochastic synchronization effect is found to be driven by the intrinsic hierarchy of neural timescales of the brain and the heterogeneous complex topology of the connectome. Moreover, the effect coincides with clustering of node phases and node frequencies and strengthening of the functional connectivity of some of the connectome's subnetworks. Overall, the work provides broad theoretical insights into the emergence and mechanisms of stochastic synchronization, highlighting its putative contribution in achieving network integration underpinning brain function.

An Overview of Emergent Order in Far-from-Equilibrium Driven Systems: From Kuramoto Oscillators to Rayleigh-Bénard Convection.

Soft-matter systems when driven out of equilibrium often give rise to structures that usually lie in between the macroscopic scale of the material and microscopic scale of its constituents. In this paper we review three such systems, the two-dimensional square-lattice Ising model, the Kuramoto model and the Rayleigh-Bénard convection system which when driven out of equilibrium give rise to emergent spatio-temporal order through self-organization. A common feature of these systems is that the entities that self-organize are coupled to one another in some way, either through local interactions or through a continuous media. Therefore, the general nature of non-equilibrium fluctuations of the intrinsic variables in these systems are found to follow similar trends as order emerges. Through this paper, we attempt to find connections between these systems, and systems in general which give rise to emergent order when driven out of equilibrium. This study, thus acts as a foundation for modeling a complex system as a two-state system, where the states: order and disorder can coexist as the system is driven away from equilibrium.

Encoding seasonal information in a two-oscillator model of the multi-oscillator circadian clock.

The suprachiasmatic nucleus (SCN) is a collection of about 10 000 neurons, each of which functions as a circadian clock with slightly different periods and phases, that work in concert with form and maintain the master circadian clock for the organism. The diversity among neurons confers on the SCN the ability to robustly encode both the 24-h light pattern as well as the seasonal time. Cluster synchronization brings the different neurons into line and reduces the large population to essentially two oscillators, coordinated by a macroscopic network motif of asymmetric repulsive-attractive coupling. We recount the steps leading to this simplification and rigorously examine the two-oscillator case by seeking an analytical solution. Through these steps, we identify physiologically relevant parameters that shape the behaviour of the SCN network and delineate its ability to store past details of seasonal variation in photoperiod.

Quantitative evaluation of simulated functional brain networks in graph theoretical analysis.

There is increasing interest in the potential of whole-brain computational models to provide mechanistic insights into resting-state brain networks. It is therefore important to determine the degree to which computational models reproduce the topological features of empirical functional brain networks. We used empirical connectivity data derived from diffusion spectrum and resting-state functional magnetic resonance imaging data from healthy individuals. Empirical and simulated functional networks, constrained by structural connectivity, were defined based on 66 brain anatomical regions (nodes). Simulated functional data were generated using the Kuramoto model in which each anatomical region acts as a phase oscillator. Network topology was studied using graph theory in the empirical and simulated data. The difference (relative error) between graph theory measures derived from empirical and simulated data was then estimated. We found that simulated data can be used with confidence to model graph measures of global network organization at different dynamic states and highlight the sensitive dependence of the solutions obtained in simulated data on the specified connection densities. This study provides a method for the quantitative evaluation and external validation of graph theory metrics derived from simulated data that can be used to inform future study designs.

Mapping how local perturbations influence systems-level brain dynamics.

The human brain exhibits a distinct spatiotemporal organization that supports brain function and can be manipulated via local brain stimulation. Such perturbations to local cortical dynamics are globally integrated by distinct neural systems. However, it remains unclear how local changes in neural activity affect large-scale system dynamics. Here, we briefly review empirical and computational studies addressing how localized perturbations affect brain activity. We then systematically analyze a model of large-scale brain dynamics, assessing how localized changes in brain activity at the different sites affect whole-brain dynamics. We find that local stimulation induces changes in brain activity that can be summarized by relatively smooth tuning curves, which relate a region's effectiveness as a stimulation site to its position within the cortical hierarchy. Our results also support the notion that brain hubs, operating in a slower regime, are more resilient to focal perturbations and critically contribute to maintain stability in global brain dynamics. In contrast, perturbations of peripheral regions, characterized by faster activity, have greater impact on functional connectivity. As a parallel with this region-level result, we also find that peripheral systems such as the visual and sensorimotor networks were more affected by local perturbations than high-level systems such as the cingulo-opercular network. Our findings highlight the importance of a periphery-to-core hierarchy to determine the effect of local stimulation on the brain network. This study also provides novel resources to orient empirical work aiming at manipulating functional connectivity using non-invasive brain stimulation.

Multiple-region directed functional connectivity based on phase delays.

Network analysis is increasingly advancing the field of neuroimaging. Neural networks are generally constructed from pairwise interactions with an assumption of linear relations between them. Here, a high-order statistical framework to calculate directed functional connectivity among multiple regions, using wavelet analysis and spectral coherence has been presented. The mathematical expression for 4 regions was derived and used to characterize a quartet of regions as a linear, combined (nonlinear), or disconnected network. Phase delays between regions were used to obtain network's temporal hierarchy and directionality. The validity of the mathematical derivation along with the effects of coupling strength and noise on its outcomes were studied by computer simulations of the Kuramoto model. The simulations demonstrated correct directionality for a large range of coupling strength and low sensitivity to Gaussian noise compared with pairwise coherences. The analysis was applied to resting-state fMRI data of 40 healthy young subjects to characterize the ventral visual system, motor system and default mode network (DMN). It was shown that the ventral visual system was predominantly composed of linear networks while the motor system and the DMN were composed of combined (nonlinear) networks. The ventral visual system exhibits its known temporal hierarchy, the motor system exhibits center ↔ out hierarchy and the DMN has dorsal ↔ ventral and anterior ↔ posterior organizations. The analysis can be applied in different disciplines such as seismology, or economy and in a variety of brain data including stimulus-driven fMRI, electrophysiology, EEG, and MEG, thus open new horizons in brain research. Hum Brain Mapp 38:1374-1386, 2017. © 2016 Wiley Periodicals, Inc.

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.

Effects of lesions on synchrony and metastability in cortical networks.

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

Electroencephalogram-Based Metastability in Mild Cognitive Impairment Alzheimer's Disease.

Background: In this study, we analyze metastability, a feature of brain dynamics in subjects experiencing mild cognitive impairment Alzheimer's disease (MCI-AD) under eyes open (EO) and eyes closed (EC) conditions. Alzheimer's disease (AD) is a critically prolonged brain disorder that interrupts neural synchronization and desynchronization. Thus, studying metastability under EO and EC conditions would help in understanding the cortical dynamics and its impact in early-stage AD. Methods: Metastability is investigated using three methods namely frequency variance analysis, Kuramoto order parameter, and through meta-state activation patterns. Frequency variance estimated from 21 electroencephalogram (EEG) channels was clustered into three regions namely anterior, central, and posterior to study the regional metastability analysis. Global metastability was assessed from Kuramoto order parameter and meta-state activation patterns by collating the 21 EEG channels. Results: Reduction in metastability was observed in central regions of MCI-AD subjects through the study of frequency variance analysis. There was a marked reduction in global metastability in the patient group under the resting EO condition. Reduction in meta-state activation properties such as temporal activation sequence complexity, modularity, and leap size in MCI-AD condition under the EO condition indicates an overall reduction in brain flexibility. Conclusion: Taken together, the study infers an underlying structural change in neuronal dynamics influencing the reduction of metastability under the MCI-AD condition. The study further revealed that this reduction in metastability is more pronounced in the EO condition.

Metastability indexes global changes in the dynamic working point of the brain following brain stimulation.

Several studies have shown that coordination among neural ensembles is a key to understand human cognition. A well charted path is to identify coordination states associated with cognitive functions from spectral changes in the oscillations of EEG or MEG. A growing number of studies suggest that the tendency to switch between coordination states, sculpts the dynamic repertoire of the brain and can be indexed by a measure known as metastability. In this article, we characterize perturbations in the metastability of global brain network dynamics following Transcranial Magnetic Stimulation that could quantify the duration for which information processing is altered. Thus allowing researchers to understand the network effects of brain stimulation, standardize stimulation protocols and design experimental tasks. We demonstrate the effect empirically using publicly available datasets and use a digital twin (a whole brain connectome model) to understand the dynamic principles that generate such observations. We observed a significant reduction in metastability, concurrent with an increase in coherence following single-pulse TMS reflecting the existence of a window where neural coordination is altered. The reduction in complexity was validated by an additional measure based on the Lempel-Ziv complexity of microstate labeled EEG data. Interestingly, higher frequencies in the EEG signal showed faster recovery in metastability than lower frequencies. The digital twin shed light on how the phase resetting introduced by the single-pulse TMS in local cortical networks can propagate globally, giving rise to changes in metastability and coherence.

Corrigendum: Study on characteristic of epileptic multi-electroencephalograph base on Hilbert-Huang transform and brain network dynamics.

[This corrects the article DOI: 10.3389/fnins.2023.1117340.].

Study on characteristic of epileptic multi-electroencephalograph base on Hilbert-Huang transform and brain network dynamics.

Lots of studies have been carried out on characteristic of epileptic Electroencephalograph (EEG). However, traditional EEG characteristic research methods lack exploration of spatial information. To study the characteristics of epileptic EEG signals from the perspective of the whole brain，this paper proposed combination methods of multi-channel characteristics from time-frequency and spatial domains. This paper was from two aspects: Firstly, signals were converted into 2D Hilbert Spectrum (HS) images which reflected the time-frequency characteristics by Hilbert-Huang Transform (HHT). These images were identified by Convolutional Neural Network (CNN) model whose sensitivity was 99.8%, accuracy was 98.7%, specificity was 97.4%, F1-score was 98.7%, and AUC-ROC was 99.9%. Secondly, the multi-channel signals were converted into brain networks which reflected the spatial characteristics by Symbolic Transfer Entropy (STE) among different channels EEG. And the results show that there are different network properties between ictal and interictal phase and the signals during the ictal enter the synchronization state more quickly, which was verified by Kuramoto model. To summarize, our results show that there was different characteristics among channels for the ictal and interictal phase, which can provide effective physical non-invasive indicators for the identification and prediction of epileptic seizures.

Reduced Plasticity in Coupling Strength in the Aging SCN Clock as Revealed by Kuramoto Modeling.

The mammalian circadian clock is located in the suprachiasmatic nucleus (SCN) and consists of a network of coupled neurons, which are entrained to the environmental light-dark cycle. The phase coherence of the neurons is plastic and driven by the duration of daylight. With aging, the capacity to behaviorally adapt to seasonal changes in photoperiod reduces. The mechanisms underlying photoperiodic adaptation are largely unknown, but are important to unravel for the development of novel interventions to improve the quality of life of the elderly. We analyzed the phase coherence of single-cell PERIOD2::LUCIFERASE (PER2::LUC) expression rhythms in the SCN of young and old mice entrained to either long or short photoperiod. The phase coherence was used as input to a 2-community noisy Kuramoto model to estimate the coupling strength between and within neuronal subpopulations. The model revealed a correlation between coupling strength and photoperiod-induced changes in the phase relationship among neurons, suggesting a functional link. We found that the SCN of young mice adapts in coupling strength over a large range, with weak coupling in long photoperiod (LP) and strong coupling in short photoperiod (SP). In aged mice, we also found weak coupling in LP, but a reduced capacity to reach strong coupling in SP. The inability to respond with an increase in coupling strength suggests that manipulation of photoperiod is not a suitable strategy to enhance clock function with aging. We conclude that the inability of aged mice to reach strong coupling contributes to deficits in behavioral adaptation to seasonal changes in photoperiod.

Weak synchronization can alter circadian period length: implications for aging and disease conditions.

The synchronization of multiple oscillators serves as the central mechanism for maintaining stable circadian rhythms in physiology and behavior. Aging and disease can disrupt synchronization, leading to changes in the periodicity of circadian activities. While our understanding of the circadian clock under synchronization has advanced significantly, less is known about its behavior outside synchronization, which can also fall within a predictable domain. These states not only impact the stability of the rhythms but also modulate the period length. In C57BL/6 mice, aging, diseases, and removal of peripheral circadian oscillators often result in lengthened behavioral circadian periods. Here, we show that these changes can be explained by a surprisingly simple mathematical relationship: the frequency is the reciprocal of the period, and its distribution becomes skewed when the period distribution is symmetric. The synchronized frequency of a population in the skewed distribution and the macroscopic frequency of combined oscillators differ, accounting for some of the atypical circadian period outputs observed in networks without synchronization. Building on this finding, we investigate the dynamics of circadian outputs in the context of aging and disease, where synchronization is weakened.