Bernadette C. M. van Wijk

Comparing brain networks of different size and connectivity density using graph theory

Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others.

Neural synchrony within the motor system: what have we learned so far?

Synchronization of neural activity is considered essential for information processing in the nervous system. Both local and inter-regional synchronization are omnipresent in different frequency regimes and relate to a variety of behavioral and cognitive functions. Over the years, many studies have sought to elucidate the question how alpha/mu, beta, and gamma synchronization contribute to motor control. Here, we review these studies with the purpose to delineate what they have added to our understanding of the neural control of movement. We highlight important findings regarding oscillations in primary motor cortex, synchronization between cortex and spinal cord, synchronization between cortical regions, as well as abnormal synchronization patterns in a selection of motor dysfunctions. The interpretation of synchronization patterns benefits from combining results of invasive and non-invasive recordings, different data analysis tools, and modeling work. Importantly, although synchronization is deemed to play a vital role, it is not the only mechanism for neural communication. Spike timing and rate coding act together during motor control and should therefore both be accounted for when interpreting movement-related activity.

Bayesian model reduction and empirical Bayes for group (DCM) studies.

This technical note describes some Bayesian procedures for the analysis of group studies that use nonlinear models at the first (within-subject) level – e.g., dynamic causal models – and linear models at subsequent (between-subject) levels. Its focus is on using Bayesian model reduction to finesse the inversion of multiple models of a single dataset or a single (hierarchical or empirical Bayes) model of multiple datasets. These applications of Bayesian model reduction allow one to consider parametric random effects and make inferences about group effects very efficiently (in a few seconds). We provide the relatively straightforward theoretical background to these procedures and illustrate their application using a worked example. This example uses a simulated mismatch negativity study of schizophrenia. We illustrate the robustness of Bayesian model reduction to violations of the (commonly used) Laplace assumption in dynamic causal modelling and show how its recursive application can facilitate both classical and Bayesian inference about group differences. Finally, we consider the application of these empirical Bayesian procedures to classification and prediction.

On the Influence of Amplitude on the Connectivity between Phases

In recent studies, functional connectivities have been reported to display characteristics of complex networks that have been suggested to concur with those of the underlying structural, i.e., anatomical, networks. Do functional networks always agree with structural ones? In all generality, this question can be answered with “no”: for instance, a fully synchronized state would imply isotropic homogeneous functional connections irrespective of the “real” underlying structure. A proper inference of structure from function and vice versa requires more than a sole focus on phase synchronization. We show that functional connectivity critically depends on amplitude variations, which implies that, in general, phase patterns should be analyzed in conjunction with the corresponding amplitude. We discuss this issue by comparing the phase synchronization patterns of interconnected Wilson–Cowan models vis-à-vis Kuramoto networks of phase oscillators. For the interconnected Wilson–Cowan models we derive analytically how connectivity between phases explicitly depends on the generating oscillators’ amplitudes. In consequence, the link between neurophysiological studies and computational models always requires the incorporation of the amplitude dynamics. Supplementing synchronization characteristics by amplitude patterns, as captured by, e.g., spectral power in M/EEG recordings, will certainly aid our understanding of the relation between structural and functional organizations in neural networks at large.

Subthalamic nucleus phase-amplitude coupling correlates with motor impairment in Parkinson's disease

Highlights • We obtained invasive subthalamic nucleus recordings in 33 Parkinson’s disease patients.• Phase–amplitude coupling between beta band and high-frequency oscillations correlates with severity of motor impairments.• Parkinsonian pathophysiology is more closely linked with low-beta band frequencies.

Granger causality revisited

This technical paper offers a critical re-evaluation of (spectral) Granger causality measures in the analysis of biological timeseries. Using realistic (neural mass) models of coupled neuronal dynamics, we evaluate the robustness of parametric and nonparametric Granger causality. Starting from a broad class of generative (state-space) models of neuronal dynamics, we show how their Volterra kernels prescribe the second-order statistics of their response to random fluctuations; characterised in terms of cross-spectral density, cross-covariance, autoregressive coefficients and directed transfer functions. These quantities in turn specify Granger causality — providing a direct (analytic) link between the parameters of a generative model and the expected Granger causality. We use this link to show that Granger causality measures based upon autoregressive models can become unreliable when the underlying dynamics is dominated by slow (unstable) modes — as quantified by the principal Lyapunov exponent. However, nonparametric measures based on causal spectral factors are robust to dynamical instability. We then demonstrate how both parametric and nonparametric spectral causality measures can become unreliable in the presence of measurement noise. Finally, we show that this problem can be finessed by deriving spectral causality measures from Volterra kernels, estimated using dynamic causal modelling.

Corticomuscular and bilateral EMG coherence reflect distinct aspects of neural synchronization.

Using electroencephalography (EEG) and electromyography (EMG), corticomuscular and bilateral motor unit synchronization have been found in different frequency bands and under different task conditions. These different types of long-range synchrony are hypothesized to originate from distinct mechanisms. We tested this by comparing time-resolved EEG-EMG and EMG-EMG coherence in a bilateral precision-grip task. Bilateral EMG activity was synchronized between 7 and 13Hz for about 1s when force output from both hands changed from an increasing to a stable force production. In contrast, EEG-EMG coherence was statistically significant between 15 and 30Hz during stable force production. The disparities in their time-frequency profiles accord with the existence of distinct underlying processes for corticomuscular and bilateral motor unit synchronization. In addition, the absence of synchronization between cortical activity and common spinal input at 10Hz renders a cortical source unlikely.

Movement-related beta oscillations show high intra-individual reliability

ABSTRACT Oscillatory activity in the beta frequency range (15–30 Hz) recorded from human sensorimotor cortex is of increasing interest as a putative biomarker of motor system function and dysfunction. Despite its increasing use in basic and clinical research, surprisingly little is known about the test‐retest reliability of spectral power and peak frequency measures of beta oscillatory signals from sensorimotor cortex. Establishing that these beta measures are stable over time in healthy populations is a necessary precursor to their use in the clinic. Here, we used scalp electroencephalography (EEG) to evaluate intra‐individual reliability of beta‐band oscillations over six sessions, focusing on changes in beta activity during movement (Movement‐Related Beta Desynchronization, MRBD) and after movement termination (Post‐Movement Beta Rebound, PMBR). Subjects performed visually‐cued unimanual wrist flexion and extension. We assessed Intraclass Correlation Coefficients (ICC) and between‐session correlations for spectral power and peak frequency measures of movement‐related and resting beta activity. Movement‐related and resting beta power from both sensorimotor cortices was highly reliable across sessions. Resting beta power yielded highest reliability (average ICC=0.903), followed by MRBD (average ICC=0.886) and PMBR (average ICC=0.663). Notably, peak frequency measures yielded lower ICC values compared to the assessment of spectral power, particularly for movement‐related beta activity (ICC=0.386–0.402). Our data highlight that power measures of movement‐related beta oscillations are highly reliable, while corresponding peak frequency measures show greater intra‐individual variability across sessions. Importantly, our finding that beta power estimates show high intra‐individual reliability over time serves to validate the notion that these measures reflect meaningful individual differences that can be utilised in basic research and clinical studies. HIGHLIGHTSMovement‐related beta‐band activity shows high test‐retest reliability.Spectral power measures are more reliable than the corresponding peak frequencies.Peak frequency measures at rest display higher reliability than during movement

Low-beta cortico-pallidal coherence decreases during movement and correlates with overall reaction time

Abstract Beta band oscillations (13–30 Hz) are a hallmark of cortical and subcortical structures that are part of the motor system. In addition to local population activity, oscillations also provide a means for synchronization of activity between regions. Here we examined the role of beta band coherence between the internal globus pallidus (GPi) and (motor) cortex during a simple reaction time task performed by nine patients with idiopathic dystonia. We recorded local field potentials from deep brain stimulation (DBS) electrodes implanted in bilateral GPi in combination with simultaneous whole‐head magneto‐encephalography (MEG). Patients responded to visually presented go or stop‐signal cues by pressing a button with left or right hand. Although coherence between signals from DBS electrodes and MEG sensors was observed throughout the entire beta band, a significant movement‐related decrease prevailed in lower beta frequencies (˜13–21 Hz). In addition, patients’ absolute coherence values in this frequency range significantly correlated with their median reaction time during the task (r = 0.89, p = 0.003). These findings corroborate the recent idea of two functionally distinct frequency ranges within the beta band, as well as the anti‐kinetic character of beta oscillations. HighlightsSimultaneous internal pallidum LFP and MEG recordings in dystonia patients.Cortico‐pallidal coherence was found throughout the beta frequency range.Predominantly low‐beta coherence (13–21 Hz) decreased with movement.Overall level of coherence was indicative of subjects median reaction time.No correlations were found between beta coherence measures and clinical scores.

Thalamo-cortical cross-frequency coupling detected with MEG.

Neural oscillations are observed across a variety of temporal and spatial scales, and are believed to play a key role in brain function. In addition to specific functional roles ascribed to isolated frequencies, it is likely that the interplay between activity at different frequencies plays an essential role in cognition and behavior (Canolty and Knight, 2010). As such, cross-frequency interactions have become the focus of a large literature exploring different types of coupling across a multitude of brain regions and states (Jirsa and Muller, 2013). Coupling between low and high frequency signal components may occur in different forms involving either the phase, amplitude or frequency of the signals. The best studied form is phase-amplitude coupling, in which the amplitude of higher frequency activity is modulated with the phase of activity in a lower frequency band. Most work to date has, however, focused on local cross-frequency interactions, leaving coupling between brain regions relatively little explored. These non-local interactions are interesting because they provide a means by which neuronal activity can be coordinated across both space and time. The paper by Roux et al. (2013) stands out because it addresses this issue non-invasively in humans. Using magnetoencephalography (MEG), Roux et al. (2013) analyzed data collected from 45 healthy participants during quiet wakefulness with eyes shut. Time series for left and right thalamus, and all other cortical and subcortical locations, were extracted by projecting band-pass filtered data through beamformer spatial filters. The instantaneous phase of thalamic alpha band activity (8–13 Hz) and the instantaneous amplitude of cortical gamma activity (30–70) were used to test for phase-amplitude coupling by computing the modulation index (Tort et al., 2008). This revealed a significant thalamo-cortical cross-frequency coupling with the posterior medial parietal cortex (PMPC), one of the hubs in the default mode network. These results are exciting, but raise an obvious concern. The thalamus is deeply located close to the center of the head and it can, therefore, be doubted whether its signal-to-noise ratio is sufficient for reliable alpha phase estimation. Furthermore, spurious thalamo-cortical coupling might have been introduced by leakage from sources elsewhere in the brain. It is vital to rule this out, and in general assess the plausibility of estimating thalamo-cortical coupling using MEG. Recently, Attal and Schwartz (2013) showed that significant thalamic sources could be detected when contrasting alpha power between eyes open and eyes closed. In addition, thalamo-cortical alpha band synchronization has been reported previously (Pollok et al., 2005). The prima facie plausibility of cross-frequency coupling between subcortical structures and the cortex on the other hand, is demonstrated by studies reporting thalamo-cortical phase-amplitude coupling using data collected invasively from the thalamus (Fitzgerald et al., 2013) and subthalamic nucleus (de Hemptinne et al., 2013). Taken together, existing literature provides some support for both the approach pursued by Roux et al. (2013) and the findings they present, however without obviating the need for careful supplementary analysis to support their findings. To this end, the authors report three extra analyses. Firstly, the transfer entropy between the raw signals from thalamus and cortex was estimated at different time lags. This indicated that information flow was directed from thalamus to cortex, and was maximal at a lag in line with physiologically plausible conduction delays. Secondly, to verify that coupling was specific to the thalamus, the authors took the gamma amplitude at PMPC as a seed and tested for coupling with the phase at all other source locations. Significant coupling was confined to the thalamus and did not appear in other regions except that gamma amplitude was also significantly coupled to alpha phase within the PMPC. Finally, correlations between spatial filters at the thalamus and all other locations in the brain supported the absence of leakage from signals at cortical locations into those of the thalamus. These analyses mitigate concerns that thalamo-cortical coupling resulted from volume conduction artefacts although further analyses could have been performed to establish this more definitively. In particular, as the authors also report phase locking between the thalamus and PMPC within the alpha band, a non-zero phase delay for this locking would argue against volume conduction. An alternative approach would have been to orthogonalize the thalamic and cortical time series as demonstrated by Hipp et al. (2012). Here, before analysing dependencies between two signals all zero-phase correlated activity is regressed out of one signal, ensuring that the time series no longer share any activity picked up from common sources. From a broader perspective, a key question is how best to understand different types of cross-frequency coupling functionally and neurophysiologically. One possibility is that activity at the frequencies of interest plays clearly defined functional roles as a pair of oscillations, such as segregating and maintaining neuronal representations of different items in working memory (Lisman and Jensen, 2013). However, an equally plausible alternative is that cross-frequency coupling is a data feature that arises as an epiphenomenon due to particular neuronal firing patterns without functional significance in itself. This would not decrease the interest of results like those reported by Roux et al. (2013), but simply change their interpretation. This discussion motivates the use of neurobiologically plausible models as a means to understand the generative mechanisms behind observed activity. Such models could include a detailed description of the neural dynamics of the different layers within a cortical column (Bastos et al., 2012). Cross-frequency phenomena might emerge by linking the layer-specific intrinsic firing frequencies (Spaak et al., 2012). It may turn out to be possible to explain complex changes in cross-frequency coupling patterns by adjusting a single parameter in the underlying neuronal model. This approach could also, in principle, support deductions about the causal relationships between sources underlying the observed coupling. Understanding interactions between neuronal dynamics at different spatial and temporal scales, both within and between regions, is likely to be critical for characterizing normal and pathological brain function. The paper of Roux et al. (2013) represents a step toward this, and suggests that MEG can play a useful role in characterizing cross-frequency coupling between deep sources and the neocortex.