Reliability of dynamic causal modelling of resting-state magnetoencephalography.

This study assesses the reliability of resting-state dynamic causal modelling (DCM) of magnetoencephalography (MEG) under conductance-based canonical microcircuit models, in terms of both posterior parameter estimates and model evidence. We use resting-state MEG data from two sessions, acquired 2 weeks apart, from a cohort with high between-subject variance arising from Alzheimer's disease. Our focus is not on the effect of disease, but on the reliability of the methods (as within-subject between-session agreement), which is crucial for future studies of disease progression and drug intervention. To assess the reliability of first-level DCMs, we compare model evidence associated with the covariance among subject-specific free energies (i.e., the 'quality' of the models) with versus without interclass correlations. We then used parametric empirical Bayes (PEB) to investigate the differences between the inferred DCM parameter probability distributions at the between subject level. Specifically, we examined the evidence for or against parameter differences (i) within-subject, within-session, and between-epochs; (ii) within-subject between-session; and (iii) within-site between-subjects, accommodating the conditional dependency among parameter estimates. We show that for data acquired close in time, and under similar circumstances, more than 95% of inferred DCM parameters are unlikely to differ, speaking to mutual predictability over sessions. Using PEB, we show a reciprocal relationship between a conventional definition of 'reliability' and the conditional dependency among inferred model parameters. Our analyses confirm the reliability and reproducibility of the conductance-based DCMs for resting-state neurophysiological data. In this respect, the implicit generative modelling is suitable for interventional and longitudinal studies of neurological and psychiatric disorders.

Multiscale Modes of Functional Brain Connectivity.

Information processing in the brain spans from localised sensorimotor processes to higher-level cognition that integrates across multiple regions. Interactions between and within these subsystems enable multiscale information processing. Despite this multiscale characteristic, functional brain connectivity is often either estimated based on 10-30 distributed modes or parcellations with 100-1000 localised parcels, both missing across-scale functional interactions. We present Multiscale Probabilistic Functional Modes (mPFMs), a new mapping which comprises modes over various scales of granularity, thus enabling direct estimation of functional connectivity within- and across-scales. Crucially, mPFMs emerged from data-driven multilevel Bayesian modelling of large functional MRI (fMRI) populations. We demonstrate that mPFMs capture both distributed brain modes and their co-existing subcomponents. In addition to validating mPFMs using simulations and real data, we show that mPFMs can predict ~900 personalised traits from UK Biobank more accurately than current standard techniques. Therefore, mPFMs can offer a paradigm shift in functional connectivity modelling and yield enhanced fMRI biomarkers for traits and diseases.

The cortical neurophysiological signature of amyotrophic lateral sclerosis.

The progressive loss of motor function characteristic of amyotrophic lateral sclerosis is associated with widespread cortical pathology extending beyond primary motor regions. Increasing muscle weakness reflects a dynamic, variably compensated brain network disorder. In the quest for biomarkers to accelerate therapeutic assessment, the high temporal resolution of magnetoencephalography is uniquely able to non-invasively capture micro-magnetic fields generated by neuronal activity across the entire cortex simultaneously. This study examined task-free magnetoencephalography to characterize the cortical oscillatory signature of amyotrophic lateral sclerosis for having potential as a pharmacodynamic biomarker. Eight to ten minutes of magnetoencephalography in the task-free, eyes-open state was recorded in amyotrophic lateral sclerosis (n = 36) and healthy age-matched controls (n = 51), followed by a structural MRI scan for co-registration. Extracted magnetoencephalography metrics from the delta, theta, alpha, beta, low-gamma, high-gamma frequency bands included oscillatory power (regional activity), 1/f exponent (complexity) and amplitude envelope correlation (connectivity). Groups were compared using a permutation-based general linear model with correction for multiple comparisons and confounders. To test whether the extracted metrics could predict disease severity, a random forest regression model was trained and evaluated using nested leave-one-out cross-validation. Amyotrophic lateral sclerosis was characterized by reduced sensorimotor beta band and increased high-gamma band power. Within the premotor cortex, increased disability was associated with a reduced 1/f exponent. Increased disability was more widely associated with increased global connectivity in the delta, theta and high-gamma bands. Intra-hemispherically, increased disability scores were particularly associated with increases in temporal connectivity and inter-hemispherically with increases in frontal and occipital connectivity. The random forest model achieved a coefficient of determination (R2) of 0.24. The combined reduction in cortical sensorimotor beta and rise in gamma power is compatible with the established hypothesis of loss of inhibitory, GABAergic interneuronal circuits in pathogenesis. A lower 1/f exponent potentially reflects a more excitable cortex and a pathology unique to amyotrophic lateral sclerosis when considered with the findings published in other neurodegenerative disorders. Power and complexity changes corroborate with the results from paired-pulse transcranial magnetic stimulation. Increased magnetoencephalography connectivity in worsening disability is thought to represent compensatory responses to a failing motor system. Restoration of cortical beta and gamma band power has significant potential to be tested in an experimental medicine setting. Magnetoencephalography-based measures have potential as sensitive outcome measures of therapeutic benefit in drug trials and may have a wider diagnostic value with further study, including as predictive markers in asymptomatic carriers of disease-causing genetic variants.

Post-task responses following working memory and movement are driven by transient spectral bursts with similar characteristics.

The post-movement beta rebound has been studied extensively using magnetoencephalography (MEG) and is reliably modulated by various task parameters as well as illness. Our recent study showed that rebounds, which we generalise as "post-task responses" (PTRs), are a ubiquitous phenomenon in the brain, occurring across the cortex in theta, alpha, and beta bands. Currently, it is unknown whether PTRs following working memory are driven by transient bursts, which are moments of short-lived high amplitude activity, similar to those that drive the post-movement beta rebound. Here, we use three-state univariate hidden Markov models (HMMs), which can identify bursts without a priori knowledge of frequency content or response timings, to compare bursts that drive PTRs in working memory and visuomotor MEG datasets. Our results show that PTRs across working memory and visuomotor tasks are driven by pan-spectral transient bursts. These bursts have very similar spectral content variation over the cortex, correlating strongly between the two tasks in the alpha (R2 = .89) and beta (R2 = .53) bands. Bursts also have similar variation in duration over the cortex (e.g., long duration bursts occur in the motor cortex for both tasks), strongly correlating over cortical regions between tasks (R2 = .56), with a mean over all regions of around 300 ms in both datasets. Finally, we demonstrate the ability of HMMs to isolate signals of interest in MEG data, such that the HMM probability timecourse correlates more strongly with reaction times than frequency filtered power envelopes from the same brain regions. Overall, we show that induced PTRs across different tasks are driven by bursts with similar characteristics, which can be identified using HMMs. Given the similarity between bursts across tasks, we suggest that PTRs across the cortex may be driven by a common underlying neural phenomenon.

Post-stroke upper limb recovery is correlated with dynamic resting-state network connectivity.

Motor recovery is still limited for people with stroke especially those with greater functional impairments. In order to improve outcome, we need to understand more about the mechanisms underpinning recovery. Task-unbiased, blood flow-independent post-stroke neural activity can be acquired from resting brain electrophysiological recordings and offers substantial promise to investigate physiological mechanisms, but behaviourally relevant features of resting-state sensorimotor network dynamics have not yet been identified. Thirty-seven people with subcortical ischaemic stroke and unilateral hand paresis of any degree were longitudinally evaluated at 3 weeks (early subacute) and 12 weeks (late subacute) after stroke. Resting-state magnetoencephalography and clinical scores of motor function were recorded and compared with matched controls. Magnetoencephalography data were decomposed using a data-driven hidden Markov model into 10 time-varying resting-state networks. People with stroke showed statistically significantly improved Action Research Arm Test and Fugl-Meyer upper extremity scores between 3 weeks and 12 weeks after stroke (both P < 0.001). Hidden Markov model analysis revealed a primarily alpha-band ipsilesional resting-state sensorimotor network which had a significantly increased life-time (the average time elapsed between entering and exiting the network) and fractional occupancy (the occupied percentage among all networks) at 3 weeks after stroke when compared with controls. The life-time of the ipsilesional resting-state sensorimotor network positively correlated with concurrent motor scores in people with stroke who had not fully recovered. Specifically, this relationship was observed only in ipsilesional rather in contralesional sensorimotor network, default mode network or visual network. The ipsilesional sensorimotor network metrics were not significantly different from controls at 12 weeks after stroke. The increased recruitment of alpha-band ipsilesional resting-state sensorimotor network at subacute stroke served as functionally correlated biomarkers exclusively in people with stroke with not fully recovered hand paresis, plausibly reflecting functional motor recovery processes.

osl-dynamics, a toolbox for modeling fast dynamic brain activity.

Neural activity contains rich spatiotemporal structure that corresponds to cognition. This includes oscillatory bursting and dynamic activity that span across networks of brain regions, all of which can occur on timescales of tens of milliseconds. While these processes can be accessed through brain recordings and imaging, modeling them presents methodological challenges due to their fast and transient nature. Furthermore, the exact timing and duration of interesting cognitive events are often a priori unknown. Here, we present the OHBA Software Library Dynamics Toolbox (osl-dynamics), a Python-based package that can identify and describe recurrent dynamics in functional neuroimaging data on timescales as fast as tens of milliseconds. At its core are machine learning generative models that are able to adapt to the data and learn the timing, as well as the spatial and spectral characteristics, of brain activity with few assumptions. osl-dynamics incorporates state-of-the-art approaches that can be, and have been, used to elucidate brain dynamics in a wide range of data types, including magneto/electroencephalography, functional magnetic resonance imaging, invasive local field potential recordings, and electrocorticography. It also provides novel summary measures of brain dynamics that can be used to inform our understanding of cognition, behavior, and disease. We hope osl-dynamics will further our understanding of brain function, through its ability to enhance the modeling of fast dynamic processes.

Evaluating functional brain organization in individuals and identifying contributions to network overlap.

Individual differences in the spatial organization of resting state networks have received increased attention in recent years. Measures of individual-specific spatial organization of brain networks and overlapping network organization have been linked to important behavioral and clinical traits and are therefore potential biomarker targets for personalized psychiatry approaches. To better understand individual-specific spatial brain organization, this paper addressed three key goals. First, we determined whether it is possible to reliably estimate weighted (non-binarized) resting state network maps using data from only a single individual, while also maintaining maximum spatial correspondence across individuals. Second, we determined the degree of spatial overlap between distinct networks, using test-retest and twin data. Third, we systematically tested multiple hypotheses (spatial mixing, temporal switching, and coupling) as candidate explanations for why networks overlap spatially. To estimate weighted network organization, we adopt the Probabilistic Functional Modes (PROFUMO) algorithm, which implements a Bayesian framework with hemodynamic and connectivity priors to supplement optimization for spatial sparsity/independence. Our findings showed that replicable individual-specific estimates of weighted resting state networks can be derived using high quality fMRI data within individual subjects. Network organization estimates using only data from each individual subject closely resembled group-informed network estimates (which was not explicitly modeled in our individual-specific analyses), suggesting that cross-subject correspondence was largely maintained. Furthermore, our results confirmed the presence of spatial overlap in network organization, which was replicable across sessions within individuals and in monozygotic twin pairs. Intriguingly, our findings provide evidence that network overlap is indicative of linear additive coupling. These results suggest that regions of network overlap concurrently process information from all contributing networks, potentially pointing to the role of overlapping network organization in the integration of information across multiple brain systems.

Interpretable many-class decoding for MEG.

Multivariate pattern analysis (MVPA) of Magnetoencephalography (MEG) and Electroencephalography (EEG) data is a valuable tool for understanding how the brain represents and discriminates between different stimuli. Identifying the spatial and temporal signatures of stimuli is typically a crucial output of these analyses. Such analyses are mainly performed using linear, pairwise, sliding window decoding models. These allow for relative ease of interpretation, e.g. by estimating a time-course of decoding accuracy, but have limited decoding performance. On the other hand, full epoch multiclass decoding models, commonly used for brain-computer interface (BCI) applications, can provide better decoding performance. However interpretation methods for such models have been designed with a low number of classes in mind. In this paper, we propose an approach that combines a multiclass, full epoch decoding model with supervised dimensionality reduction, while still being able to reveal the contributions of spatiotemporal and spectral features using permutation feature importance. Crucially, we introduce a way of doing supervised dimensionality reduction of input features within a neural network optimised for the classification task, improving performance substantially. We demonstrate the approach on 3 different many-class task-MEG datasets using image presentations. Our results demonstrate that this approach consistently achieves higher accuracy than the peak accuracy of a sliding window decoder while estimating the relevant spatiotemporal features in the MEG signal.

A data-driven network decomposition of the temporal, spatial, and spectral dynamics underpinning visual-verbal working memory processes.

The brain dynamics underlying working memory (WM) unroll via transient frequency-specific large-scale brain networks. This multidimensionality (time, space, and frequency) challenges traditional analyses. Through an unsupervised technique, the time delay embedded-hidden Markov model (TDE-HMM), we pursue a functional network analysis of magnetoencephalographic data from 38 healthy subjects acquired during an n-back task. Here we show that this model inferred task-specific networks with unique temporal (activation), spectral (phase-coupling connections), and spatial (power spectral density distribution) profiles. A theta frontoparietal network exerts attentional control and encodes the stimulus, an alpha temporo-occipital network rehearses the verbal information, and a broad-band frontoparietal network with a P300-like temporal profile leads the retrieval process and motor response. Therefore, this work provides a unified and integrated description of the multidimensional working memory dynamics that can be interpreted within the neuropsychological multi-component model of WM, improving the overall neurophysiological and neuropsychological comprehension of WM functioning.

Group-level brain decoding with deep learning.

Decoding brain imaging data are gaining popularity, with applications in brain-computer interfaces and the study of neural representations. Decoding is typically subject-specific and does not generalise well over subjects, due to high amounts of between subject variability. Techniques that overcome this will not only provide richer neuroscientific insights but also make it possible for group-level models to outperform subject-specific models. Here, we propose a method that uses subject embedding, analogous to word embedding in natural language processing, to learn and exploit the structure in between-subject variability as part of a decoding model, our adaptation of the WaveNet architecture for classification. We apply this to magnetoencephalography data, where 15 subjects viewed 118 different images, with 30 examples per image; to classify images using the entire 1 s window following image presentation. We show that the combination of deep learning and subject embedding is crucial to closing the performance gap between subject- and group-level decoding models. Importantly, group models outperform subject models on low-accuracy subjects (although slightly impair high-accuracy subjects) and can be helpful for initialising subject models. While we have not generally found group-level models to perform better than subject-level models, the performance of group modelling is expected to be even higher with bigger datasets. In order to provide physiological interpretation at the group level, we make use of permutation feature importance. This provides insights into the spatiotemporal and spectral information encoded in the models. All code is available on GitHub (https://github.com/ricsinaruto/MEG-group-decode).

Dissecting unsupervised learning through hidden Markov modeling in electrophysiological data.

Unsupervised, data-driven methods are commonly used in neuroscience to automatically decompose data into interpretable patterns. These patterns differ from one another depending on the assumptions of the models. How these assumptions affect specific data decompositions in practice, however, is often unclear, which hinders model applicability and interpretability. For instance, the hidden Markov model (HMM) automatically detects characteristic, recurring activity patterns (so-called states) from time series data. States are defined by a certain probability distribution, whose state-specific parameters are estimated from the data. But what specific features, from all of those that the data contain, do the states capture? That depends on the choice of probability distribution and on other model hyperparameters. Using both synthetic and real data, we aim to better characterize the behavior of two HMM types that can be applied to electrophysiological data. Specifically, we study which differences in data features (such as frequency, amplitude, or signal-to-noise ratio) are more salient to the models and therefore more likely to drive the state decomposition. Overall, we aim at providing guidance for the appropriate use of this type of analysis on one- or two-channel neural electrophysiological data and an informed interpretation of its results given the characteristics of the data and the purpose of the analysis.NEW & NOTEWORTHY Compared with classical supervised methods, unsupervised methods of analysis have the advantage to be freer of subjective biases. However, it is not always clear what aspects of the data these methods are most sensitive to, which complicates interpretation. Focusing on the hidden Markov model, commonly used to describe electrophysiological data, we explore in detail the nature of its estimates through simulations and real data examples, providing important insights about what to expect from these models.

Multi-modal and multi-model interrogation of large-scale functional brain networks.

Existing whole-brain models are generally tailored to the modelling of a particular data modality (e.g., fMRI or MEG/EEG). We propose that despite the differing aspects of neural activity each modality captures, they originate from shared network dynamics. Building on the universal principles of self-organising delay-coupled nonlinear systems, we aim to link distinct features of brain activity - captured across modalities - to the dynamics unfolding on a macroscopic structural connectome. To jointly predict connectivity, spatiotemporal and transient features of distinct signal modalities, we consider two large-scale models - the Stuart Landau and Wilson and Cowan models - which generate short-lived 40 Hz oscillations with varying levels of realism. To this end, we measure features of functional connectivity and metastable oscillatory modes (MOMs) in fMRI and MEG signals - and compare them against simulated data. We show that both models can represent MEG functional connectivity (FC), functional connectivity dynamics (FCD) and generate MOMs to a comparable degree. This is achieved by adjusting the global coupling and mean conduction time delay and, in the WC model, through the inclusion of balance between excitation and inhibition. For both models, the omission of delays dramatically decreased the performance. For fMRI, the SL model performed worse for FCD and MOMs, highlighting the importance of balanced dynamics for the emergence of spatiotemporal and transient patterns of ultra-slow dynamics. Notably, optimal working points varied across modalities and no model was able to achieve a correlation with empirical FC higher than 0.4 across modalities for the same set of parameters. Nonetheless, both displayed the emergence of FC patterns that extended beyond the constraints of the anatomical structure. Finally, we show that both models can generate MOMs with empirical-like properties such as size (number of brain regions engaging in a mode) and duration (continuous time interval during which a mode appears). Our results demonstrate the emergence of static and dynamic properties of neural activity at different timescales from networks of delay-coupled oscillators at 40 Hz. Given the higher dependence of simulated FC on the underlying structural connectivity, we suggest that mesoscale heterogeneities in neural circuitry may be critical for the emergence of parallel cross-modal functional networks and should be accounted for in future modelling endeavours.

Reduced coupling between offline neural replay events and default mode network activation in schizophrenia.

Schizophrenia is characterized by an abnormal resting state and default mode network brain activity. However, despite intense study, the mechanisms linking default mode network dynamics to neural computation remain elusive. During rest, sequential hippocampal reactivations, known as 'replay', are played out within default mode network activation windows, highlighting a potential role of replay-default mode network coupling in memory consolidation and model-based mental simulation. Here, we test a hypothesis of reduced replay-default mode network coupling in schizophrenia, using magnetoencephalography and a non-spatial sequence learning task designed to elicit off-task (i.e. resting state) neural replay. Participants with a diagnosis of schizophrenia (n = 28, mean age 28.2 years, range 20-40, 6 females, 13 not taking antipsychotic medication) and non-clinical control participants (n = 29, mean age 28.1 years, range 18-45, 6 females, matched at group level for age, intelligence quotient, gender, years in education and working memory) underwent a magnetoencephalography scan both during task completion and during a post-task resting state session. We used neural decoding to infer the time course of default mode network activation (time-delay embedding hidden Markov model) and spontaneous neural replay (temporally delayed linear modelling) in resting state magnetoencephalography data. Using multiple regression, we then quantified the extent to which default mode network activation was uniquely predicted by replay events that recapitulated the learned task sequences (i.e. 'task-relevant' replay-default mode network coupling). In control participants, replay-default mode network coupling was augmented following sequence learning, an augmentation that was specific for replay of task-relevant (i.e. learned) state transitions. This task-relevant replay-default mode network coupling effect was significantly reduced in schizophrenia (t(52) = 3.93, P = 0.018). Task-relevant replay-default mode network coupling predicted memory maintenance of learned sequences (ρ(52) = 0.31, P = 0.02). Importantly, reduced task-relevant replay-default mode network coupling in schizophrenia was not explained by differential replay or altered default mode network dynamics between groups nor by reference to antipsychotic exposure. Finally, task-relevant replay-default mode network coupling during rest correlated with stimulus-evoked default mode network modulation as measured in a separate task session. In the context of a proposed functional role of replay-default mode network coupling, our findings shed light on the functional significance of default mode network abnormalities in schizophrenia and provide for a consilience between task-based and resting state default mode network findings in this disorder.

Mapping Interictal activity in epilepsy using a hidden Markov model: A magnetoencephalography study.

Epilepsy is a highly heterogeneous neurological disorder with variable etiology, manifestation, and response to treatment. It is imperative that new models of epileptiform brain activity account for this variability, to identify individual needs and allow clinicians to curate personalized care. Here, we use a hidden Markov model (HMM) to create a unique statistical model of interictal brain activity for 10 pediatric patients. We use magnetoencephalography (MEG) data acquired as part of standard clinical care for patients at the Children's Hospital of Philadelphia. These data are routinely analyzed using excess kurtosis mapping (EKM); however, as cases become more complex (extreme multifocal and/or polymorphic activity), they become harder to interpret with EKM. We assessed the performance of the HMM against EKM for three patient groups, with increasingly complicated presentation. The difference in localization of epileptogenic foci for the two methods was 7 ± 2 mm (mean ± SD over all 10 patients); and 94% ± 13% of EKM temporal markers were matched by an HMM state visit. The HMM localizes epileptogenic areas (in agreement with EKM) and provides additional information about the relationship between those areas. A key advantage over current methods is that the HMM is a data-driven model, so the output is tuned to each individual. Finally, the model output is intuitive, allowing a user (clinician) to review the result and manually select the HMM epileptiform state, offering multiple advantages over previous methods and allowing for broader implementation of MEG epileptiform analysis in surgical decision-making for patients with intractable epilepsy.

New Therapeutics in Alzheimer's Disease Longitudinal Cohort study (NTAD): study protocol.

INTRODUCTION: With the pressing need to develop treatments that slow or stop the progression of Alzheimer's disease, new tools are needed to reduce clinical trial duration and validate new targets for human therapeutics. Such tools could be derived from neurophysiological measurements of disease. METHODS AND ANALYSIS: The New Therapeutics in Alzheimer's Disease study (NTAD) aims to identify a biomarker set from magneto/electroencephalography that is sensitive to disease and progression over 1 year. The study will recruit 100 people with amyloid-positive mild cognitive impairment or early-stage Alzheimer's disease and 30 healthy controls aged between 50 and 85 years. Measurements of the clinical, cognitive and imaging data (magnetoencephalography, electroencephalography and MRI) of all participants will be taken at baseline. These measurements will be repeated after approximately 1 year on participants with Alzheimer's disease or mild cognitive impairment, and clinical and cognitive assessment of these participants will be repeated again after approximately 2 years. To assess reliability of magneto/electroencephalographic changes, a subset of 30 participants with mild cognitive impairment or early-stage Alzheimer's disease will also undergo repeat magneto/electroencephalography 2 weeks after baseline. Baseline and longitudinal changes in neurophysiology are the primary analyses of interest. Additional outputs will include atrophy and cognitive change and estimated numbers needed to treat each arm of simulated clinical trials of a future disease-modifying therapy. ETHICS AND DATA STATEMENT: The study has received a favourable opinion from the East of England Cambridge Central Research Ethics Committee (REC reference 18/EE/0042). Results will be disseminated through internal reports, peer-reviewed scientific journals, conference presentations, website publication, submission to regulatory authorities and other publications. Data will be made available via the Dementias Platform UK Data Portal on completion of initial analyses by the NTAD study group.

Modulations of local synchrony over time lead to resting-state functional connectivity in a parsimonious large-scale brain model.

Biophysical models of large-scale brain activity are a fundamental tool for understanding the mechanisms underlying the patterns observed with neuroimaging. These models combine a macroscopic description of the within- and between-ensemble dynamics of neurons within a single architecture. A challenge for these models is accounting for modulations of within-ensemble synchrony over time. Such modulations in local synchrony are fundamental for modeling behavioral tasks and resting-state activity. Another challenge comes from the difficulty in parametrizing large scale brain models which hinders researching principles related with between-ensembles differences. Here we derive a parsimonious large scale brain model that can describe fluctuations of local synchrony. Crucially, we do not reduce within-ensemble dynamics to macroscopic variables first, instead we consider within and between-ensemble interactions similarly while preserving their physiological differences. The dynamics of within-ensemble synchrony can be tuned with a parameter which manipulates local connectivity strength. We simulated resting-state static and time-resolved functional connectivity of alpha band envelopes in models with identical and dissimilar local connectivities. We show that functional connectivity emerges when there are high fluctuations of local and global synchrony simultaneously (i.e. metastable dynamics). We also show that for most ensembles, leaning towards local asynchrony or synchrony correlates with the functional connectivity with other ensembles, with the exception of some regions belonging to the default-mode network.

Understanding Harmonic Structures Through Instantaneous Frequency.

The analysis of harmonics and non-sinusoidal waveform shape in time-series data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an integer frequency ratio, constant phase, and a well-defined joint instantaneous frequency. We show this definition is linked to extrema counting and Empirical Mode Decomposition (EMD). We explore the mathematics of our definition and link it to results from analytic number theory. This naturally leads to us to define two classes of harmonic structures, termed strong and weak, with different extrema behaviour. We validate our framework using both simulations and real data. Specifically, we look at the harmonic structures in shallow water waves, the FitzHugh-Nagumo neuronal model, and the non-sinusoidal theta oscillation in rat hippocampus local field potential data. We further discuss how our definition helps to address mode splitting in nonlinear time-series decomposition methods. A clear understanding of when harmonics are present in signals will enable a deeper understanding of the functional roles of non-sinusoidal oscillations.

Mixtures of large-scale dynamic functional brain network modes.

Accurate temporal modelling of functional brain networks is essential in the quest for understanding how such networks facilitate cognition. Researchers are beginning to adopt time-varying analyses for electrophysiological data that capture highly dynamic processes on the order of milliseconds. Typically, these approaches, such as clustering of functional connectivity profiles and Hidden Markov Modelling (HMM), assume mutual exclusivity of networks over time. Whilst a powerful constraint, this assumption may be compromising the ability of these approaches to describe the data effectively. Here, we propose a new generative model for functional connectivity as a time-varying linear mixture of spatially distributed statistical "modes". The temporal evolution of this mixture is governed by a recurrent neural network, which enables the model to generate data with a rich temporal structure. We use a Bayesian framework known as amortised variational inference to learn model parameters from observed data. We call the approach DyNeMo (for Dynamic Network Modes), and show using simulations it outperforms the HMM when the assumption of mutual exclusivity is violated. In resting-state MEG, DyNeMo reveals a mixture of modes that activate on fast time scales of 100-150 ms, which is similar to state lifetimes found using an HMM. In task MEG data, DyNeMo finds modes with plausible, task-dependent evoked responses without any knowledge of the task timings. Overall, DyNeMo provides decompositions that are an approximate remapping of the HMM's while showing improvements in overall explanatory power. However, the magnitude of the improvements suggests that the HMM's assumption of mutual exclusivity can be reasonable in practice. Nonetheless, DyNeMo provides a flexible framework for implementing and assessing future modelling developments.

The relationship between frequency content and representational dynamics in the decoding of neurophysiological data.

Decoding of high temporal resolution, stimulus-evoked neurophysiological data is increasingly used to test theories about how the brain processes information. However, a fundamental relationship between the frequency spectra of the neural signal and the subsequent decoding accuracy timecourse is not widely recognised. We show that, in commonly used instantaneous signal decoding paradigms, each sinusoidal component of the evoked response is translated to double its original frequency in the subsequent decoding accuracy timecourses. We therefore recommend, where researchers use instantaneous signal decoding paradigms, that more aggressive low pass filtering is applied with a cut-off at one quarter of the sampling rate, to eliminate representational alias artefacts. However, this does not negate the accompanying interpretational challenges. We show that these can be resolved by decoding paradigms that utilise both a signal's instantaneous magnitude and its local gradient information as features for decoding. On a publicly available MEG dataset, this results in decoding accuracy metrics that are higher, more stable over time, and free of the technical and interpretational challenges previously characterised. We anticipate that a broader awareness of these fundamental relationships will enable stronger interpretations of decoding results by linking them more clearly to the underlying signal characteristics that drive them.

Transient beta activity and cortico-muscular connectivity during sustained motor behaviour.

Neural oscillations are thought to play a central role in orchestrating activity states between distant neural populations. For example, during isometric contraction, 13-30 Hz beta activity becomes phase coupled between the motor cortex and the contralateral muscle. This and related observations have led to the proposal that beta activity and connectivity sustain stable cognitive and motor states - or the 'status quo' - in the brain. Recently, however, beta activity at the single-trial level has been shown to be short-lived - though so far this has been reported for regional beta activity in tasks without sustained motor demands. Here, we measured magnetoencephalography (MEG) and electromyography (EMG) in 18 human participants performing a sustained isometric contraction (gripping) task. If cortico-muscular beta connectivity is directly responsible for sustaining a stable motor state, then beta activity within single trials should be (or become) sustained in this context. In contrast, we found that motor beta activity and connectivity with the downstream muscle were transient. Moreover, we found that sustained motor requirements did not prolong beta-event duration in comparison to rest. These findings suggest that neural synchronisation between the brain and the muscle involves short 'bursts' of frequency-specific connectivity, even when task demands - and motor behaviour - are sustained.