Inference of time-evolving coupled dynamical systems in the presence of noise

Effect of Sequential Repetitive Transcranial Magnetic Stimulation With Bilateral Arm Training on the Brain Effective Connectivity in Chronic Stroke

This study investigated the effects of combining repetitive transcranial magnetic stimulation (rTMS) with bilateral arm training (BAT) on effective brain connectivity in chronic stroke patients using functional near-infrared spectroscopy. Fifteen post-stroke patients and fifteen healthy individuals were enrolled. Coupling function analysis was performed to evaluate the effective connectivity inflow, outflow, and the dominant information flow (DIF) during standalone BAT and combined rTMS-BAT therapy. Significant task-related alterations were observed in the ipsilesional supplementary motor area and occipital lobe (OL) of stroke patients undergoing rTMS-BAT. During BAT, stroke patients exhibited more pronounced DIF from the OL to motor areas compared to healthy controls. Furthermore, in the rTMS-BAT condition, patients demonstrated enhanced DIF from the ipsilesional OL and contralesional prefrontal cortex to ipsilesional motor areas. These findings suggested a potential synergistic effect on cortical reorganization through the sequential combination of task-related training and TMS in chronic stroke patients, offering insights into rehabilitation strategies.

Methamphetamine alters the circadian oscillator and its couplings on multiple scales in Per1/2/3 knockout mice

Disruptions to circadian rhythms in mammals are associated with alterations in their physiological and mental states. Circadian rhythms are currently analyzed in the time domain using approaches such as actograms, thus failing to appreciate their time-localized characteristics, time-varying nature and multiscale dynamics. In this study, we apply time-resolved analysis to investigate behavioral rhythms in Per1/2/3 knockout (KO) mice and their changes following methamphetamine administration, focusing on circadian (around 24 h), low-frequency ultradian (around 7 h), high-frequency ultradian (around 30 min), and circabidian (around 48 h) oscillations. In the absence of methamphetamine, Per1/2/3 KO mice in constant darkness exhibited a dominant, ∼7 h oscillation. We demonstrate that methamphetamine exposure restores the circadian rhythm, although the frequency of the methamphetamine sensitive circadian oscillator varied considerably compared to the highly regular wild-type circadian rhythm. Additionally, methamphetamine increased multiscale activity and induced a circabidian oscillation in the Per1/2/3 KO mice. The information transfer between oscillatory modes, with frequencies around circadian, low-frequency ultradian and high-frequency ultradian activity, due to their mutual couplings, was also investigated. For Per1/2/3 KO mice in constant darkness, the most prevalent coupling was between low and high-frequency ultradian activity. Following methamphetamine administration, the coupling between the circadian and high-frequency ultradian activity became dominant. In each case, the direction of information transfer was between the corresponding phases from the slower to faster oscillations. The time-varying nature of the circadian rhythm exhibited in the absence of Per1/2/3 genes and following methamphetamine administration may have profound implications for health and disease.

Enhancing cardiovascular monitoring: a non-linear model for characterizing RR interval fluctuations in exercise and recovery

This work aimed to develop and validate a novel non-linear model to characterize RR interval (RRi) time-dependent fluctuations throughout a rest-exercise-recovery protocol, offering a more precise and physiologically relevant representation of cardiac autonomic responses than traditional HRV metrics or linear approaches. Using data from a cohort of 272 elderly participants, the model employs logistic functions to capture the non-stationary and transient nature of RRi time-dependent fluctuations, with parameter estimation achieved via Hamiltonian Monte Carlo. Sobol sensitivity analysis identified baseline RRi (α) and recovery proportion (c) as the primary drivers of variability, underscoring their critical roles in autonomic regulation and resilience. Validation against real-world RRi data demonstrated robust model performance (R2 = 0.868, CI95%[0.834, 0.895] and Root Mean Square Error [RMSE] = 32.6 ms, CI95%[30.01, 35.77]), accurately reflecting autonomic recovery and exercise-induced fluctuations. By advancing real-time cardiovascular assessments, this framework holds significant potential for clinical applications in rehabilitation and cardiovascular monitoring in athletic contexts to optimize performance and recovery. These findings highlight the model's ability to provide precise, physiologically relevant assessments of autonomic function, paving the way for its use in personalized health monitoring and performance optimization across diverse populations.

Theta and alpha connectivity in children with autism spectrum disorder

Spontaneous electroencephalography (EEG) measurements have demonstrated putative variations in the neural connectivity of subjects with autism spectrum disorder, as compared to neurotypical individuals. However, the exact nature of these connectivity differences has remained unknown, a question that we now address. Resting-state, eyes-open EEG data were recorded over 20 min from a cohort of 13 males aged 3-5 years with autism spectrum disorder, and nine neurotypical individuals as a control group. We use time-localized, phase-based methods of data analysis, including wavelet phase coherence and dynamical Bayesian inference. Several 3 min signal segments were analysed to evaluate the reproducibility of the proposed measures. In the autism spectrum disorder cohort, we demonstrate a significant (P < 0.05) reduction in functional connectivity strength across all frontal probe pairs. In addition, the percentage of time during which frontal regions were coupled was significantly reduced in the autism spectrum disorder group compared to the control group. These changes remained consistent across repeated measurements. To further validate the findings, an additional resting-state EEG dataset (eyes open and closed) from 67 individuals with autism spectrum disorder and 66 control group individuals (male, 5-15 years) was assessed. The functional connectivity results demonstrated a reduction in theta and alpha connectivity on a local, but not global, level. No association was found with age. The connectivity differences observed suggest the potential of theta and alpha connectivity as biomarkers for autism spectrum disorder. Additionally, the robustness to amplitude perturbations of the methods proposed here makes them particularly suitable for the clinical assessment of autism spectrum disorder and of the efficacy of therapeutic interventions.

Cortico-Muscular Phase Connectivity During an Isometric Knee Extension Task in People with Early Parkinson's Disease

INTRODUCTION

Parkinson's disease (PD) is characterized by enhanced beta-band activity (13-30 Hz) in the motor control regions. Simultaneously, cortico-muscular (CM) connectivity in the beta-band during iso-metric contractions tends to decline with age, in various diseases, and under dual-task conditions.

OBJECTIVE

This study aimed to characterize electroencephalograph (EEG) and electromyograph (EMG) power spectra during a motor task, assess CM phase connectivity, and explore how these measures are modulated by an additional cognitive task. Specifically, we focused on the beta-band to explore the relationship between heightened beta amplitude and reduced beta CM connectivity.

METHODOLOGY

Early-stage people with PD and age-matched controls performed an isometric knee extension task, a cognitive task, and a combined dual task, while EEG (128ch) and EMG (2x32ch) were recorded. CM phase connectivity was assessed through phase coherence and a phase dynamics model.

RESULTS

The EEG power spectrum revealed no cohort differences in the beta-band. EMG also showed no differences up to 80 Hz. However, the combined EEG-EMG analysis uncovered reduced beta phase coherence in people with early PD during the motor task. CM phase coherence exhibited distinct scalp topography and frequency ranges compared to the EEG power spectrum, suggesting different mechanisms for pathological beta increase and CM connectivity. Additionally, phase dynamics modelling indicated stronger directional coupling from the cortex to the active muscle and less prominent phase coupling across people with PD. Despite high inter-individual variability, these metrics may prove useful for personalized assessments, particularly in people with heightened CM connectivity.

Setting of the Poincaré section for accurately calculating the phase of rhythmic spatiotemporal dynamics

Synchronization analysis of real-world systems is essential across numerous fields, including physics, chemistry, and life sciences. Generally, the governing equations of these systems are unknown, and thus, the phase is calculated from measurements. Although existing phase calculation techniques are designed for oscillators that possess no spatial structure, methods for handling spatiotemporal dynamics remain undeveloped. The presence of spatial structure complicates the determination of which measurements should be used for accurate phase calculation. To address this, we explore a method for calculating the phase from measurements taken at a single spatial grid point. The phase is calculated to increase linearly between event times when the measurement time series intersects the Poincaré section. The difference between the calculated phase and the isochron-based phase, resulting from the discrepancy between the isochron and the Poincaré section, is evaluated using a linear approximation near the limit-cycle solution. We found that the difference is small when measurements are taken from regions that dominate the rhythms of the entire spatiotemporal dynamics. Furthermore, we investigate an alternative method where the Poincaré section is applied to time series obtained through orthogonal decomposition of the entire spatiotemporal dynamics. We present two decomposition schemes that utilize principal component analysis. For illustration, the phase is calculated from the measurements of spatiotemporal dynamics exhibiting target waves or oscillating spots, simulated by weakly coupled FitzHugh-Nagumo reaction-diffusion models.

Characterization of cardiorespiratory coupling via a variability-based multi-method approach: Application to postural orthostatic tachycardia syndrome

There are several mechanisms responsible for the dynamical link between heart period (HP) and respiration (R), usually referred to as cardiorespiratory coupling (CRC). Historically, diverse signal processing techniques have been employed to study CRC from the spontaneous fluctuations of HP and respiration (R). The proposed tools differ in terms of rationale and implementation, capturing diverse aspects of CRC. In this review, we classify the existing methods and stress differences with the aim of proposing a variability-based multi-method approach to CRC evaluation. Ten methodologies for CRC estimation, namely, power spectral decomposition, traditional and causal squared coherence,\;information transfer, cross-conditional entropy, mixed prediction, Shannon entropy of the latency between heartbeat and inspiratory/expiratory onset, conditional entropy of the phase dynamics, synchrogram-based analysis, pulse-respiration quotient, and joint symbolic dynamics, are considered. The ability of these techniques was exemplified over recordings acquired from patients suffering from postural orthostatic tachycardia syndrome (POTS) and healthy controls. Analyses were performed at rest in the supine position (REST) and during head-up tilt (HUT). Although most of the methods indicated that at REST, the CRC was lower in POTS patients and decreased more evidently during HUT in POTS, peculiar differences stressed the complementary value of the approaches. The multiple perspectives provided by the variability-based multi-method approach to CRC evaluation help the characterization of a pathological state and/or the quantification of the effect of a postural challenge. The present work stresses the need for the application of multiple methods to derive a more complete evaluation of the CRC in humans.

Reconstruction of phase dynamics from macroscopic observations based on linear and nonlinear response theories

We propose a method to reconstruct the phase dynamics in rhythmical interacting systems from macroscopic responses to weak inputs by developing linear and nonlinear response theories, which predict the responses in a given system. By solving an inverse problem, the method infers an unknown system: the natural frequency distribution, the coupling function, and the time delay which is inevitable in real systems. In contrast to previous methods, our method requires neither strong invasiveness nor microscopic observations. We demonstrate that the method reconstructs two phase systems from observed responses accurately. The qualitative methodological advantages demonstrated by our quantitative numerical examinations suggest its broad applicability in various fields, including brain systems, which are often observed through macroscopic signals such as electroencephalograms and functional magnetic response imaging.

Trained recurrent neural networks develop phase-locked limit cycles in a working memory task

Neural oscillations are ubiquitously observed in many brain areas. One proposed functional role of these oscillations is that they serve as an internal clock, or 'frame of reference'. Information can be encoded by the timing of neural activity relative to the phase of such oscillations. In line with this hypothesis, there have been multiple empirical observations of such phase codes in the brain. Here we ask: What kind of neural dynamics support phase coding of information with neural oscillations? We tackled this question by analyzing recurrent neural networks (RNNs) that were trained on a working memory task. The networks were given access to an external reference oscillation and tasked to produce an oscillation, such that the phase difference between the reference and output oscillation maintains the identity of transient stimuli. We found that networks converged to stable oscillatory dynamics. Reverse engineering these networks revealed that each phase-coded memory corresponds to a separate limit cycle attractor. We characterized how the stability of the attractor dynamics depends on both reference oscillation amplitude and frequency, properties that can be experimentally observed. To understand the connectivity structures that underlie these dynamics, we showed that trained networks can be described as two phase-coupled oscillators. Using this insight, we condensed our trained networks to a reduced model consisting of two functional modules: One that generates an oscillation and one that implements a coupling function between the internal oscillation and external reference. In summary, by reverse engineering the dynamics and connectivity of trained RNNs, we propose a mechanism by which neural networks can harness reference oscillations for working memory. Specifically, we propose that a phase-coding network generates autonomous oscillations which it couples to an external reference oscillation in a multi-stable fashion.

Blood pressure-cerebral oxygen coupling model: A new approach for stroke risk prediction

Stroke is a major cause of death and disability worldwide, but predicting its risk remains challenging. This study aimed to evaluate the cerebral blood flow autoregulation function of subjects with different stroke risk levels and predict their stroke risk. The coupling strength between cerebral oxygen and blood pressure signals was calculated by wavelet analysis and dynamic Bayesian inference and used as a quantitative index of cerebral blood flow autoregulation. A stroke prediction model based on the extreme random tree was constructed using the coupling strength and other data as input features. The results showed that the coupling strength was significantly higher in the high-risk group than the other groups. Moreover, the prediction model achieved an average accuracy of 0.80 across the three groups. The coupling strength of cerebral oxygen and blood pressure can be used as an objective index to predict stroke risk, which has implications for stroke prevention and intervention.

The time-evolving epileptic brain network: concepts, definitions, accomplishments, perspectives

Epilepsy is now considered a network disease that affects the brain across multiple levels of spatial and temporal scales. The paradigm shift from an epileptic focus-a discrete cortical area from which seizures originate-to a widespread epileptic network-spanning lobes and hemispheres-considerably advanced our understanding of epilepsy and continues to influence both research and clinical treatment of this multi-faceted high-impact neurological disorder. The epileptic network, however, is not static but evolves in time which requires novel approaches for an in-depth characterization. In this review, we discuss conceptual basics of network theory and critically examine state-of-the-art recording techniques and analysis tools used to assess and characterize a time-evolving human epileptic brain network. We give an account on current shortcomings and highlight potential developments towards an improved clinical management of epilepsy.

Inferring connectivity of an oscillatory network via the phase dynamics reconstruction

We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.

Delta-alpha cross-frequency coupling for different brain regions

Neural interactions occur on different levels and scales. It is of particular importance to understand how they are distributed among different neuroanatomical and physiological relevant brain regions. We investigated neural cross-frequency couplings between different brain regions according to the Desikan-Killiany brain parcellation. The adaptive dynamic Bayesian inference method was applied to EEG measurements of healthy resting subjects in order to reconstruct the coupling functions. It was found that even after averaging over all subjects, the mean coupling function showed a characteristic waveform, confirming the direct influence of the delta-phase on the alpha-phase dynamics in certain brain regions and that the shape of the coupling function changes for different regions. While the averaged coupling function within a region was of similar form, the region-averaged coupling function was averaged out, which implies that there is a common dependence within separate regions across the subjects. It was also found that for certain regions the influence of delta on alpha oscillations is more pronounced and that oscillations that influence other are more evenly distributed across brain regions than the influenced oscillations. When presenting the information on brain lobes, it was shown that the influence of delta emanating from the brain as a whole is greatest on the alpha oscillations of the cingulate frontal lobe, and at the same time the influence of delta from the cingulate parietal brain lobe is greatest on the alpha oscillations of the whole brain.

Neural Cross-Frequency Coupling Functions in Sleep

The human brain presents a heavily connected complex system. From a relatively fixed anatomy, it can enable a vast repertoire of functions. One important brain function is the process of natural sleep, which alters consciousness and voluntary muscle activity. On neural level, these alterations are accompanied by changes of the brain connectivity. In order to reveal the changes of connectivity associated with sleep, we present a methodological framework for reconstruction and assessment of functional interaction mechanisms. By analyzing EEG (electroencephalogram) recordings from human whole night sleep, first, we applied a time-frequency wavelet transform to study the existence and strength of brainwave oscillations. Then we applied a dynamical Bayesian inference on the phase dynamics in the presence of noise. With this method we reconstructed the cross-frequency coupling functions, which revealed the mechanism of how the interactions occur and manifest. We focus our analysis on the delta-alpha coupling function and observe how this cross-frequency coupling changes during the different sleep stages. The results demonstrated that the delta-alpha coupling function was increasing gradually from Awake to NREM3 (non-rapid eye movement), but only during NREM2 and NREM3 deep sleep it was significant in respect of surrogate data testing. The analysis on the spatially distributed connections showed that this significance is strong only for within the single electrode region and in the front-to-back direction. The presented methodological framework is for the whole-night sleep recordings, but it also carries general implications for other global neural states.

An extended Hilbert transform method for reconstructing the phase from an oscillatory signal

Rhythmic activity is ubiquitous in biological systems from the cellular to organism level. Reconstructing the instantaneous phase is the first step in analyzing the essential mechanism leading to a synchronization state from the observed signals. A popular method of phase reconstruction is based on the Hilbert transform, which can only reconstruct the interpretable phase from a limited class of signals, e.g., narrow band signals. To address this issue, we propose an extended Hilbert transform method that accurately reconstructs the phase from various oscillatory signals. The proposed method is developed by analyzing the reconstruction error of the Hilbert transform method with the aid of Bedrosian's theorem. We validate the proposed method using synthetic data and show its systematically improved performance compared with the conventional Hilbert transform method with respect to accurately reconstructing the phase. Finally, we demonstrate that the proposed method is potentially useful for detecting the phase shift in an observed signal. The proposed method is expected to facilitate the study of synchronization phenomena from experimental data.

Cortical response induced by task-oriented training of the upper limb in subacute stroke patients as assessed by functional near-infrared spectroscopy

Despite the popularity of task-oriented training for stroke, the cortical reorganization associated with this type of therapy remains to be fully elucidated due to the lack of dynamic assessment tools. A good tolerance for motion artifacts makes functional near-infrared spectroscopy (fNIRS) suitable for investigating task-induced cortical responses in stroke patients. Here, patients were randomly assigned to receive task oriented (n = 25) or cyclic rotary training (n = 25) with simultaneous cortical activation and effective connectivity network analysis between prefrontal and motor cortices (PFC/MC). Compared with cyclic rotary training, task-oriented training induced significantly increased activation in both hemispheres and enhanced influence of PFC on MC. In addition, significantly decreased activation lateralization and increased betweenness centrality of the contralesional MC suggested widespread involvement of the contralesional hemisphere during task-oriented training. This study verifies the feasibility of fNIRS combined with motor paradigms for assessing neural responses associated with stroke rehabilitation in real time.

Age-dependent cardiorespiratory directional coupling in wake-resting state

Objective.Cooperation in the cardiorespiratory system helps maintain internal stability. Various types of system interactions have been investigated; however, the characteristics of the interactions have mostly been studied using data collected in well-defined physiological states, such as sleep. Furthermore, most analyses provided general information about the interaction, making it difficult to quantify how the systems influenced one another.Approach.Cardiorespiratory directional coupling was investigated in different age groups (20 young and 19 elderly subjects) in a wake-resting state. The directionality index (DI) was calculated using instantaneous phases from the heartbeat interval and respiratory signal to provide information about the strength and direction of interaction between the systems. Statistical analysis was performed between the groups on the DI and independent measures of directionality (nc_r: influence from cardiac system to respiratory system, and nc_c: influence from the respiratory system to the cardiac system).Main results.The values of DI were -0.52 and -0.17 in the young and elderly groups, respectively (p< 0.001). Furthermore, the values of nc_rand nc_cwere found to be significantly different between the groups (p< 0.001), respectively.Significance.Changes in both directions between the systems influence different aspects of cardiorespiratory coupling between the groups. This observation could be linked to different levels of autonomic modulation associated with ageing. Our approach could aid in quantitatively tracking and comprehending how systems interact in response to physiological and environmental changes. It could also be used to understand how abnormal interaction characteristics influence physiological system dysfunctions and disorders.

Integrating psychosocial variables and societal diversity in epidemic models for predicting COVID-19 transmission dynamics

During the current COVID-19 pandemic, governments must make decisions based on a variety of information including estimations of infection spread, health care capacity, economic and psychosocial considerations. The disparate validity of current short-term forecasts of these factors is a major challenge to governments. By causally linking an established epidemiological spread model with dynamically evolving psychosocial variables, using Bayesian inference we estimate the strength and direction of these interactions for German and Danish data of disease spread, human mobility, and psychosocial factors based on the serial cross-sectional COVID-19 Snapshot Monitoring (COSMO; N = 16,981). We demonstrate that the strength of cumulative influence of psychosocial variables on infection rates is of a similar magnitude as the influence of physical distancing. We further show that the efficacy of political interventions to contain the disease strongly depends on societal diversity, in particular group-specific sensitivity to affective risk perception. As a consequence, the model may assist in quantifying the effect and timing of interventions, forecasting future scenarios, and differentiating the impact on diverse groups as a function of their societal organization. Importantly, the careful handling of societal factors, including support to the more vulnerable groups, adds another direct instrument to the battery of political interventions fighting epidemic spread.

Interacting Oscillators with Fluctuating Coupling: Mode Mixing without Cross-Correlations

Coupled oscillators are one of the basic models in nonlinear dynamics. Here, we study numerically and analytically the spectra of two harmonic oscillators with stochastically fluctuating coupling and driving forces reproducing a thermostat. We show that, even at small coupling, vanishing on average, the oscillation spectra exhibit mixing, even though no cross-correlations exists between the oscillators. Our results reveal a new mechanism of mode mixing for stochastically uncorrelated systems that is crucial for analysis of spectra in various systems, from simple liquids to living systems.

Normalizing the brain connectome for communication through synchronization

Networks in neuroscience determine how brain function unfolds, and their perturbations lead to psychiatric disorders and brain disease. Brain networks are characterized by their connectomes, which comprise the totality of all connections, and are commonly described by graph theory. This approach is deeply rooted in a particle view of information processing, based on the quantification of informational bits such as firing rates. Oscillations and brain rhythms demand, however, a wave perspective of information processing based on synchronization. We extend traditional graph theory to a dual, particle-wave, perspective, integrate time delays due to finite transmission speeds, and derive a normalization of the connectome. When applied to the database of the Human Connectome Project, it explains the emergence of frequency-specific network cores including the visual and default mode networks. These findings are robust across human subjects (N = 100) and are a fundamental network property within the wave picture. The normalized connectome comprises the particle view in the limit of infinite transmission speeds and opens the applicability of graph theory to a wide range of novel network phenomena, including physiological and pathological brain rhythms. These two perspectives are orthogonal, but not incommensurable, when understood within the novel, here-proposed, generalized framework of structural connectivity.

Beyond the Baroreflex: A New Measure of Autonomic Regulation Based on the Time-Frequency Assessment of Variability, Phase Coherence and Couplings

For decades the role of autonomic regulation and the baroreflex in the generation of the respiratory sinus arrhythmia (RSA) - modulation of heart rate by the frequency of breathing - has been under dispute. We hypothesized that by using autonomic blockers we can reveal which oscillations and their interactions are suppressed, elucidating their involvement in RSA as well as in cardiovascular regulation more generally. R-R intervals, end tidal CO2, finger arterial pressure, and muscle sympathetic nerve activity (MSNA) were measured simultaneously in 7 subjects during saline, atropine and propranolol infusion. The measurements were repeated during spontaneous and fixed-frequency breathing, and apnea. The power spectra, phase coherence and couplings were calculated to characterise the variability and interactions within the cardiovascular system. Atropine reduced R-R interval variability (p < 0.05) in all three breathing conditions, reduced MSNA power during apnea and removed much of the significant coherence and couplings. Propranolol had smaller effect on the power of oscillations and did not change the number of significant interactions. Most notably, atropine reduced R-R interval power in the 0.145-0.6 Hz interval during apnea, which supports the hypothesis that the RSA is modulated by a mechanism other than the baroreflex. Atropine also reduced or made negative the phase shift between the systolic and diastolic pressure, indicating the cessation of baroreflex-dependent blood pressure variability. This result suggests that coherent respiratory oscillations in the blood pressure can be used for the non-invasive assessment of autonomic regulation.

Identifying Cognitive Impairment in Elderly Using Coupling Functions Between Cerebral Oxyhemoglobin and Arterial Blood Pressure

Background

This study aimed to assess brain oxygenation status and cerebral autoregulation function in subjects with cognitive dysfunction.

Methods

The Montreal Cognitive Assessment (MoCA) was applied to divide the subjects into three groups: cognitive impairment (Group CI, 72.50 ± 10.93 y), mild cognitive impairment (Group MCI, 72.02 ± 9.90 y), and normal cognition (Group NC, 70.72 ± 7.66 y). Near-infrared spectroscopy technology and a non-invasive blood pressure device were used to simultaneously measure changes in cerebral tissue oxygenation signals in the bilateral prefrontal lobes (LPFC/RPFC) and arterial blood pressure (ABP) signals from subjects in the resting state (15 min). The coupling between ABP and cerebral oxyhemoglobin concentrations (Δ [O2Hb]) was calculated in very-low-frequency (VLF, 0.02-0.07 Hz) and low-frequency (LF, 0.07-0.2 Hz) bands based on the dynamical Bayesian inference approach. Pearson correlation analyses were used to study the relationships between MoCA scores, tissue oxygenation index, and strength of coupling function.

Results

In the interval VLF, Group CI (p = 0.001) and Group MCI (p = 0.013) exhibited significantly higher coupling strength from ABP to Δ [O2Hb] in the LPFC than Group NC. In the interval LF, coupling strength from ABP to Δ [O2Hb] in the LPFC was significantly higher in Group CI than in Group NC (p = 0.001). Pearson correlation results showed that MoCA scores had a significant positive correlation with the tissue oxygenation index and a significant negative correlation with the coupling strength from ABP to Δ [O2Hb].

Conclusion

The significantly increased coupling strength may be evidence of impaired cerebral autoregulation function in subjects with cognitive dysfunction. The Pearson correlation results suggest that indicators of brain oxygenation status and cerebral autoregulation function can reflect cognitive function. This study provides insights into the mechanisms underlying the pathophysiology of cognitive impairment and provides objective indicators for screening cognitive impairment in the elderly population.

Multiscale Time-resolved Analysis Reveals Remaining Behavioral Rhythms in Mice Without Canonical Circadian Clocks

Circadian rhythms are internal processes repeating approximately every 24 hours in living organisms. The dominant circadian pacemaker is synchronized to the environmental light-dark cycle. Other circadian pacemakers, which can have noncanonical circadian mechanisms, are revealed by arousing stimuli, such as scheduled feeding, palatable meals and running wheel access, or methamphetamine administration. Organisms also have ultradian rhythms, which have periods shorter than circadian rhythms. However, the biological mechanism, origin, and functional significance of ultradian rhythms are not well-elucidated. The dominant circadian rhythm often masks ultradian rhythms; therefore, we disabled the canonical circadian clock of mice by knocking out Per1/2/3 genes, where Per1 and Per2 are essential components of the mammalian light-sensitive circadian mechanism. Furthermore, we recorded wheel-running activity every minute under constant darkness for 272 days. We then investigated rhythmic components in the absence of external influences, applying unique multiscale time-resolved methods to analyze the oscillatory dynamics with time-varying frequencies. We found four rhythmic components with periods of ∼17 h, ∼8 h, ∼4 h, and ∼20 min. When the ∼17-h rhythm was prominent, the ∼8-h rhythm was of low amplitude. This phenomenon occurred periodically approximately every 2-3 weeks. We found that the ∼4-h and ∼20-min rhythms were harmonics of the ∼8-h rhythm. Coupling analysis of the ridge-extracted instantaneous frequencies revealed strong and stable phase coupling from the slower oscillations (∼17, ∼8, and ∼4 h) to the faster oscillations (∼20 min), and weak and less stable phase coupling in the reverse direction and between the slower oscillations. Together, this study elucidated the relationship between the oscillators in the absence of the canonical circadian clock, which is critical for understanding their functional significance. These studies are essential as disruption of circadian rhythms contributes to diseases, such as cancer and obesity, as well as mood disorders.

Blinking coupling enhances network synchronization

This paper studies the synchronization of a network with linear diffusive coupling, which blinks between the variables periodically. The synchronization of the blinking network in the case of sufficiently fast blinking is analyzed by showing that the stability of the synchronous solution depends only on the averaged coupling and not on the instantaneous coupling. To illustrate the effect of the blinking period on the network synchronization, the Hindmarsh-Rose model is used as the dynamics of nodes. The synchronization is investigated by considering constant single-variable coupling, averaged coupling, and blinking coupling through a linear stability analysis. It is observed that by decreasing the blinking period, the required coupling strength for synchrony is reduced. It equals that of the averaged coupling model times the number of variables. However, in the averaged coupling, all variables participate in the coupling, while in the blinking model only one variable is coupled at any time. Therefore, the blinking coupling leads to an enhanced synchronization in comparison with the single-variable coupling. Numerical simulations of the average synchronization error of the network confirm the results obtained from the linear stability analysis.

Linear and Nonlinear Directed Connectivity Analysis of the Cardio-Respiratory System in Type 1 Diabetes

In this study, we explored the possibility of developing non-invasive biomarkers for patients with type 1 diabetes (T1D) by quantifying the directional couplings between the cardiac, vascular, and respiratory systems, treating them as interconnected nodes in a network configuration. Towards this goal, we employed a linear directional connectivity measure, the directed transfer function (DTF), estimated by a linear multivariate autoregressive modelling of ECG, respiratory and skin perfusion signals, and a nonlinear method, the dynamical Bayesian inference (DBI) analysis of bivariate phase interactions. The physiological data were recorded concurrently for a relatively short time period (5 min) from 10 healthy control subjects and 10 T1D patients. We found that, in both control and T1D subjects, breathing had greater influence on the heart and perfusion with respect to the opposite coupling direction and that, by both employed methods of analysis, the causal influence of breathing on the heart was significantly decreased (p < 0.05) in T1D patients compared to the control group. These preliminary results, although obtained from a limited number of subjects, provide a strong indication for the usefulness of a network-based multi-modal analysis for the development of biomarkers of T1D-related complications from short-duration data, as well as their potential in the exploration of the pathophysiological mechanisms that underlie this devastating and very widespread disease.

Generative formalism of causality quantifiers for processes

The concept of dynamical causal effect (DCE) is generalized and equipped with a formalism which allows one to formulate in a unified manner and interrelate a variety of causality quantifiers used in time series analysis. An elementary DCE from a subsystem Y to a subsystem X is defined within the stochastic dynamical systems framework as a response of a future X state to an appropriate variation of an initial (X,Y)-state distribution or a certain parameter of Y or of the coupling element Y→X; this response is quantified in a probabilistic sense via a certain distinction functional; elementary DCEs are assembled over a set of initial variations via an assemblage functional. To include all those aspects, a "triple brackets formula" for the general DCE is suggested and serves as a first principle to produce specific causality quantifiers as realizations of the general DCE. As an application, transfer entropy and Liang-Kleeman information flow are related surprisingly as opposite limit cases in a family of DCEs; it is shown that their "nats per time unit" may differ drastically. The suggested DCE viewpoint links any formal causality quantifier to "intervention-effect" experiments, i.e., future responses to initial variations, and so provides its dynamical interpretation, opening a way to its further physical interpretations in studies of physical systems.

Noninvasive inference methods for interaction and noise intensities of coupled oscillators using only spike time data

Identifying interactions is essential for understanding self-organized systems because they are a source of order, function, and complexity. However, distinguishing interaction and noise effect is generally difficult because they both critically affect the stability of an ordered state. We propose methods that enable us to simultaneously infer both interaction and noise intensities. Our methods use only the time series of periodic events such as spike time data and do not require any external stimuli. Moreover, it is not necessary to assume a function form to fit. We numerically demonstrate that our methods yield reasonable inference even for a relatively short time series. These features are particularly beneficial for application in biological and chemical complex systems.

Measurements of interaction intensity are generally achieved by observing responses to perturbations. In biological and chemical systems, external stimuli tend to deteriorate their inherent nature, and thus, it is necessary to develop noninvasive inference methods. In this paper, we propose theoretical methods to infer coupling strength and noise intensity simultaneously in two well-synchronized noisy oscillators through observations of spontaneously fluctuating events such as neural spikes. A phase oscillator model is applied to derive formulae relating each of the parameters to spike time statistics. Using these formulae, each parameter is inferred from a specific set of statistics. We verify these methods using the FitzHugh–Nagumo model as well as the phase model. Our methods do not require external perturbations and thus can be applied to various experimental systems.

Average beta burst duration profiles provide a signature of dynamical changes between the ON and OFF medication states in Parkinson's disease

Parkinson's disease motor symptoms are associated with an increase in subthalamic nucleus beta band oscillatory power. However, these oscillations are phasic, and there is a growing body of evidence suggesting that beta burst duration may be of critical importance to motor symptoms. This makes insights into the dynamics of beta bursting generation valuable, in particular to refine closed-loop deep brain stimulation in Parkinson's disease. In this study, we ask the question "Can average burst duration reveal how dynamics change between the ON and OFF medication states?". Our analysis of local field potentials from the subthalamic nucleus demonstrates using linear surrogates that the system generating beta oscillations is more likely to act in a non-linear regime OFF medication and that the change in a non-linearity measure is correlated with motor impairment. In addition, we pinpoint the simplest dynamical changes that could be responsible for changes in the temporal patterning of beta oscillations between medication states by fitting to data biologically inspired models, and simpler beta envelope models. Finally, we show that the non-linearity can be directly extracted from average burst duration profiles under the assumption of constant noise in envelope models. This reveals that average burst duration profiles provide a window into burst dynamics, which may underlie the success of burst duration as a biomarker. In summary, we demonstrate a relationship between average burst duration profiles, dynamics of the system generating beta oscillations, and motor impairment, which puts us in a better position to understand the pathology and improve therapies such as deep brain stimulation.

Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing

The precise mechanisms connecting the cardiovascular system and the cerebrospinal fluid (CSF) are not well understood in detail. This paper investigates the couplings between the cardiac and respiratory components, as extracted from blood pressure (BP) signals and oscillations of the subarachnoid space width (SAS), collected during slow ventilation and ventilation against inspiration resistance. The experiment was performed on a group of 20 healthy volunteers (12 females and 8 males; BMI =22.1±3.2 kg/m2; age 25.3±7.9 years). We analysed the recorded signals with a wavelet transform. For the first time, a method based on dynamical Bayesian inference was used to detect the effective phase connectivity and the underlying coupling functions between the SAS and BP signals. There are several new findings. Slow breathing with or without resistance increases the strength of the coupling between the respiratory and cardiac components of both measured signals. We also observed increases in the strength of the coupling between the respiratory component of the BP and the cardiac component of the SAS and vice versa. Slow breathing synchronises the SAS oscillations, between the brain hemispheres. It also diminishes the similarity of the coupling between all analysed pairs of oscillators, while inspiratory resistance partially reverses this phenomenon. BP–SAS and SAS–BP interactions may reflect changes in the overall biomechanical characteristics of the brain.

Time Window Determination for Inference of Time-Varying Dynamics: Application to Cardiorespiratory Interaction

Interacting dynamical systems abound in nature, with examples ranging from biology and population dynamics, through physics and chemistry, to communications and climate. Often their states, parameters and functions are time-varying, because such systems interact with other systems and the environment, exchanging information and matter. A common problem when analysing time-series data from dynamical systems is how to determine the length of the time window for the analysis. When one needs to follow the time-variability of the dynamics, or the dynamical parameters and functions, the time window needs to be resolved first. We tackled this problem by introducing a method for adaptive determination of the time window for interacting oscillators, as modeled and scaled for the cardiorespiratory interaction. By investigating a system of coupled phase oscillators and utilizing the Dynamical Bayesian Inference method, we propose a procedure to determine the time window and the propagation parameter of the covariance matrix. The optimal values are determined so that the inferred parameters follow the dynamics of the actual ones and at the same time the error of the inference represented by the covariance matrix is minimal. The effectiveness of the methodology is presented on a system of coupled limit-cycle oscillators and on the cardiorespiratory interaction. Three cases of cardiorespiratory interaction were considered-measurement with spontaneous free breathing, one with periodic sine breathing and one with a-periodic time-varying breathing. The results showed that the cardiorespiratory coupling strength and similarity of form of coupling functions have greater values for slower breathing, and this variability follows continuously the change of the breathing frequency. The method can be applied effectively to other time-varying oscillatory interactions and carries important implications for analysis of general dynamical systems.

A pilot study: Wavelet cross-correlation of cardiovascular oscillations under controlled respiration in humans

The influence of deep controlled respiration on cardiovascular oscillations in 13 healthy young volunteers was studied. A measurement system comprising electrocardiography, laser Doppler flowmetry (LDF) and photoplethysmography (PPG) was used to estimate heart rate variability (HRV), tissue blood volume and skin blood perfusion at spontaneous respiration and during three tests at controlled conditions. In the latter case, respiration was controlled in both rate (0.04, 0.1 and 0.25 Hz) and depth. During respiration at 0.04 and 0.1 Hz, the amplification of a respiratory-related component in the spectra of HRV and PPG signals turned out to be more significant than that at spontaneous respiration, and at 0.25 Hz this component remained unchanged. Controlled respiration caused a significant increase in correlation in HRV-PPG, HRV-LDF and PPG-LDF pairs of signals compared to spontaneous one. At 0.25 Hz controlled respiration, no significant increase in correlation in these pairs of signals was found. The differences observed in this study can be attributed to the effects of the sympathetic nerve activity on vascular tone regulation.

Static and dynamic attractive-repulsive interactions in two coupled nonlinear oscillators

Many systems exhibit both attractive and repulsive types of interactions, which may be dynamic or static. A detailed understanding of the dynamical properties of a system under the influence of dynamically switching attractive or repulsive interactions is of practical significance. However, it can also be effectively modeled with two coexisting competing interactions. In this work, we investigate the effect of time-varying attractive-repulsive interactions as well as the hybrid model of coexisting attractive-repulsive interactions in two coupled nonlinear oscillators. The dynamics of two coupled nonlinear oscillators, specifically limit cycles as well as chaotic oscillators, are studied in detail for various dynamical transitions for both cases. Here, we show that dynamic or static attractive-repulsive interactions can induce an important transition from the oscillatory to steady state in identical nonlinear oscillators due to competitive effects. The analytical condition for the stable steady state in dynamic interactions at the low switching time period and static coexisting interactions are calculated using linear stability analysis, which is found to be in good agreement with the numerical results. In the case of a high switching time period, oscillations are revived for higher interaction strength.

Detecting network structures from measurable data produced by dynamics with hidden variables

Depicting network structures from measurable data is of significance. In real-world situations, it is common that some variables of networks are unavailable or even unknown. These unavailable and unknown variables, i.e., hidden variables, will lead to much reconstruction error, even make reconstruction methods useless. In this paper, to solve hidden variable problems, we propose three reconstruction methods, respectively, based on the following conditions: statistical characteristics of hidden variables, linearizable hidden variables, and white noise injection. Among them, the method based on white noise injection is active and invasive. In our framework, theoretic analyses of these three methods are given at first, and, furthermore, the validity of theoretical derivations and the robustness of these methods are fully verified through numerical results. Our work may be, therefore, helpful for practical experiments.

On the concept of dynamical reduction: the case of coupled oscillators

An overview is given on two representative methods of dynamical reduction known as centre-manifold reduction and phase reduction. These theories are presented in a somewhat more unified fashion than the theories in the past. The target systems of reduction are coupled limit-cycle oscillators. Particular emphasis is placed on the remarkable structural similarity existing between these theories. While the two basic principles, i.e. (i) reduction of dynamical degrees of freedom and (ii) transformation of reduced evolution equation to a canonical form, are shared commonly by reduction methods in general, it is shown how these principles are incorporated into the above two reduction theories in a coherent manner. Regarding the phase reduction, a new formulation of perturbative expansion is presented for discrete populations of oscillators. The style of description is intended to be so informal that one may digest, without being bothered with technicalities, what has been done after all under the word reduction. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences

Dynamical systems are widespread, with examples in physics, chemistry, biology, population dynamics, communications, climatology and social science. They are rarely isolated but generally interact with each other. These interactions can be characterized by coupling functions-which contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how each interaction occurs. Coupling functions can be used, not only to understand, but also to control and predict the outcome of the interactions. This theme issue assembles ground-breaking work on coupling functions by leading scientists. After overviewing the field and describing recent advances in the theory, it discusses novel methods for the detection and reconstruction of coupling functions from measured data. It then presents applications in chemistry, neuroscience, cardio-respiratory physiology, climate, electrical engineering and social science. Taken together, the collection summarizes earlier work on coupling functions, reviews recent developments, presents the state of the art, and looks forward to guide the future evolution of the field. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Synchronization transitions caused by time-varying coupling functions

Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase- synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

A practical method for estimating coupling functions in complex dynamical systems

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Median Nerve Electrical Stimulation-Induced Changes in Effective Connectivity in Patients With Stroke as Assessed With Functional Near-Infrared Spectroscopy

Background. The cortical plastic changes in response to median nerve electrical stimulation (MNES) in stroke patients have not been entirely illustrated. Objective. This study aimed to investigate MNES-related changes in effective connectivity (EC) within a cortical network after stroke by using functional near-infrared spectroscopy (fNIRS). Methods. The cerebral oxygenation signals in the bilateral prefrontal cortex (LPFC/RPFC), motor cortex (LMC/RMC), and occipital lobe (LOL/ROL) of 20 stroke patients with right hemiplegia were measured by fNIRS in 2 conditions: (1) resting state and (2) MNES applied to the right wrist. Coupling function together with dynamical Bayesian inference was used to assess MNES-related changes in EC among the cerebral low-frequency fluctuations. Results. Compared with the resting state, EC from LPFC and RPFC to LOL was significantly increased during the MNES state in stroke patients. Additionally, MNES triggered significantly higher coupling strengths from LMC and LOL to RPFC. The interregional main coupling direction was observed from LPFC to bilateral motor and occipital areas in responding to MNES, suggesting that MNES could promote the regulation function of ipsilesional prefrontal areas in the functional network. MNES can induce muscle twitch of the stroke-affected hand involving a decreased neural coupling of the contralesional motor area on the ipsilesional MC. Conclusions. MNES can trigger sensorimotor stimulations of the affected hand that sequentially involved functional reorganization of distant cortical areas after stroke. Investigating MNES-related changes in EC after stroke may help further our understanding of the neural mechanisms underlying MNES.

Reconstructing dynamical networks via feature ranking

Empirical data on real complex systems are becoming increasingly available. Parallel to this is the need for new methods of reconstructing (inferring) the structure of networks from time-resolved observations of their node-dynamics. The methods based on physical insights often rely on strong assumptions about the properties and dynamics of the scrutinized network. Here, we use the insights from machine learning to design a new method of network reconstruction that essentially makes no such assumptions. Specifically, we interpret the available trajectories (data) as "features" and use two independent feature ranking approaches-Random Forest and RReliefF-to rank the importance of each node for predicting the value of each other node, which yields the reconstructed adjacency matrix. We show that our method is fairly robust to coupling strength, system size, trajectory length, and noise. We also find that the reconstruction quality strongly depends on the dynamical regime.

Variable Preloading Force in an Archetypal Oscillator

In this paper, the influence of the time variable preloading force on the vibration of an archetypal oscillator is investigated. The oscillator is modeled as a slider-string system which is mathematically described with a second order nonlinear differential equation with time variable parameters. An approximate procedure for solving the equation is introduced. It is based on the exact solution of the pure nonlinear equation in the form of the Ateb function. The obtained result gives the vibration amplitude and phase variation of the oscillator depending on the preloading force variation. Based on this result, the procedure for regulation of the preloading force as the function of the required vibration amplitude decrease is developed. It is concluded that the preloading force may be used as a control parameter of the oscillator.

Limb linkage rehabilitation training-related changes in cortical activation and effective connectivity after stroke: A functional near-infrared spectroscopy study

Stroke remains the leading cause of long-term disability worldwide. Rehabilitation training is essential for motor function recovery following stroke. Specifically, limb linkage rehabilitation training can stimulate motor function in the upper and lower limbs simultaneously. This study aimed to investigate limb linkage rehabilitation task-related changes in cortical activation and effective connectivity (EC) within a functional brain network after stroke by using functional near-infrared spectroscopy (fNIRS) imaging. Thirteen stroke patients with either left hemiparesis (L-H group, n = 6) and or right hemiparesis (R-H group, n = 7) and 16 healthy individuals (control group) participated in this study. A multichannel fNIRS system was used to measure changes in cerebral oxygenated hemoglobin (delta HbO2) and deoxygenated hemoglobin (delta HHb) in the bilateral prefrontal cortices (PFCs), motor cortices (MCs), and occipital lobes (OLs) during (1) the resting state and (2) a motor rehabilitation task with upper and lower limb linkage (first 10 min [task_S1], last 10 min [task_S2]). The frequency-specific EC among the brain regions was calculated based on coupling functions and dynamic Bayesian inference in frequency intervals: high-frequency I (0.6-2 Hz) and II (0.145-0.6 Hz), low-frequency III (0.052-0.145 Hz), and very-low-frequency IV (0.021-0.052 Hz). The results showed that the stroke patients exhibited an asymmetric (greater activation in the contralesional versus ipsilesional motor region) cortical activation pattern versus healthy controls. Compared with the healthy controls, the stroke patients showed significantly lower EC (p < 0.025) in intervals I and II in the resting and task states. The EC from the MC and OL to the right PFC in interval IV was significantly higher in the R-H group than in the control group during the resting and task states (p < 0.025). Furthermore, the L-H group showed significantly higher EC from the MC and OL to the left PFC in intervals III and IV during the task states compared with the control group (p < 0.025). The significantly increased influence of the MC and OL on the contralesional PFC in low- and very-low-frequency bands suggested that plastic reorganization of cognitive resources severed to compensate for impairment in stroke patients during the motor rehabilitation task. This study can serve as a basis for understanding task-related reorganization of functional brain networks and developing novel assessment techniques for stroke rehabilitation.

Dynamic causal modelling of fluctuating connectivity in resting-state EEG

Functional and effective connectivity are known to change systematically over time. These changes might be explained by several factors, including intrinsic fluctuations in activity-dependent neuronal coupling and contextual factors, like experimental condition and time. Furthermore, contextual effects may be subject-specific or conserved over subjects. To characterize fluctuations in effective connectivity, we used dynamic causal modelling (DCM) of cross spectral responses over 1- min of electroencephalogram (EEG) recordings during rest, divided into 1-sec windows. We focused on two intrinsic networks: the default mode and the saliency network. DCM was applied to estimate connectivity in each time-window for both networks. Fluctuations in DCM connectivity parameters were assessed using hierarchical parametric empirical Bayes (PEB). Within-subject, between-window effects were modelled with a second-level linear model with temporal basis functions as regressors. This procedure was conducted for every subject separately. Bayesian model reduction was then used to assess which (combination of) temporal basis functions best explain dynamic connectivity over windows. A third (between-subject) level model was used to infer which dynamic connectivity parameters are conserved over subjects. Our results indicate that connectivity fluctuations in the default mode network and to a lesser extent the saliency network comprised both subject-specific components and a common component. For both networks, connections to higher order regions appear to monotonically increase during the 1- min period. These results not only establish the predictive validity of dynamic connectivity estimates - in virtue of detecting systematic changes over subjects - they also suggest a network-specific dissociation in the relative contribution of fluctuations in connectivity that depend upon experimental context. We envisage these procedures could be useful for characterizing brain state transitions that may be explained by their cognitive or neuropathological underpinnings.

Reconstruction of effective connectivity in the case of asymmetric phase distributions

BACKGROUND

The interaction of different brain regions is supported by transient synchronization between neural oscillations at different frequencies. Different measures based on synchronization theory are used to assess the strength of the interactions from experimental data. One method of estimating the effective connectivity between brain regions, within the framework of the theory of weakly coupled phase oscillators, was implemented in Dynamic Causal Modelling (DCM) for phase coupling (Penny et al., 2009). However, the results of such an approach strongly depend on the observables used to reconstruct the equations (Kralemann et al., 2008). In particular, an asymmetric distribution of the observables could result in a false estimation of the effective connectivity between the network nodes.

NEW METHOD

In this work we built a new modelling part into DCM for phase coupling, and extended it with a distortion function that accommodates departures from purely sinusoidal oscillations.

RESULTS

By analysing numerically generated data sets with an asymmetric phase distribution, we demonstrated that the extended DCM for phase coupling with the additional modelling component, correctly estimates the coupling functions.

COMPARISON WITH EXISTING METHODS

The new method allows for different intrinsic frequencies among coupled neuronal populations and provides results that do not depend on the distribution of the observables.

CONCLUSIONS

The proposed method can be used to analyse effective connectivity between brain regions within and between different frequency bands, to characterize m:n phase coupling, and to unravel underlying mechanisms of the transient synchronization.

Numerical phase reduction beyond the first order approximation

We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach.

The Role of Data in Model Building and Prediction: A Survey Through Examples

The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play—and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components.

Optimizing the detection of nonstationary signals by using recurrence analysis

Recurrence analysis and its quantifiers are strongly dependent on the evaluation of the vicinity threshold parameter, i.e., the threshold to regard two points close enough in phase space to be considered as just one. We develop a new way to optimize the evaluation of the vicinity threshold in order to assure a higher level of sensitivity to recurrence quantifiers to allow the detection of even small changes in the dynamics. It is used to promote recurrence analysis as a tool to detect nonstationary behavior of time signals or space profiles. We show that the ability to detect small changes provides information about the present status of the physical process responsible to generate the signal and offers mechanisms to predict future states. Here, a higher sensitive recurrence analysis is proposed as a precursor, a tool to predict near future states of a particular system, based on just (experimentally) obtained signals of some available variables of the system. Comparisons with traditional methods of recurrence analysis show that the optimization method developed here is more sensitive to small variations occurring in a signal. The method is applied to numerically generated time series as well as experimental data from physiology.

Phase-lags in large scale brain synchronization: Methodological considerations and in-silico analysis

Architecture of phase relationships among neural oscillations is central for their functional significance but has remained theoretically poorly understood. We use phenomenological model of delay-coupled oscillators with increasing degree of topological complexity to identify underlying principles by which the spatio-temporal structure of the brain governs the phase lags between oscillatory activity at distant regions. Phase relations and their regions of stability are derived and numerically confirmed for two oscillators and for networks with randomly distributed or clustered bimodal delays, as a first approximation for the brain structural connectivity. Besides in-phase, clustered delays can induce anti-phase synchronization for certain frequencies, while the sign of the lags is determined by the natural frequencies and by the inhomogeneous network interactions. For in-phase synchronization faster oscillators always phase lead, while stronger connected nodes lag behind the weaker during frequency depression, which consistently arises for in-silico results. If nodes are in anti-phase regime, then a distance π is added to the in-phase trends. The statistics of the phases is calculated from the phase locking values (PLV), as in many empirical studies, and we scrutinize the method’s impact. The choice of surrogates do not affects the mean of the observed phase lags, but higher significance levels that are generated by some surrogates, cause decreased variance and might fail to detect the generally weaker coherence of the interhemispheric links. These links are also affected by the non-stationary and intermittent synchronization, which causes multimodal phase lags that can be misleading if averaged. Taken together, the results describe quantitatively the impact of the spatio-temporal connectivity of the brain to the synchronization patterns between brain regions, and to uncover mechanisms through which the spatio-temporal structure of the brain renders phases to be distributed around 0 and π.

Trial registration: South African Clinical Trials Register: http://www.sanctr.gov.za/SAClinicalbrnbspTrials/tabid/169/Default.aspx, then link to respiratory tract then link to tuberculosis, pulmonary; and TASK Applied Sciences Clinical Trials, AP-TB-201-16 (ALOPEXX): https://task.org.za/clinical-trials/.

Functional connectivity, and in particular, phase coupling between distant brain regions may be fundamental in regulating neuronal processing and communication. However, phase relationships between the nodes of the brain and how they are confined by its spatio-temporal structure, have been mostly overlooked. We use a model of oscillatory dynamics superimposed on the space-time structure defined by the connectome, and we analyze the possible regimes of synchronization. Limitations of data analysis are also considered and we show that the choice of the significance threshold for coherence does not essentially impact the statistics of the observed phase lags, although it is crucial for the right detection of statistically significant coherence. Analytical insights are obtained for networks with heterogeneous time-delays, based on the empirical data from the connectome, and these are confirmed by numerical simulations, which show in- or anti-phase synchronization depending on the frequency and the distribution of time-delays. Phase lags are shown to result from inhomogeneous network interactions, so that stronger connected nodes generally phase lag behind the weaker.