Localized energy for wave equations with degenerate trapping

Localized energy estimates have become a fundamental tool when studying wave equations in the presence of asymptotically at background geometry. Trapped rays necessitate a loss when compared to the estimate on Minkowski space. A loss of regularity is a common way to incorporate such. When trapping is sufficiently weak, a logarithmic loss of regularity suffices. Here, by studying a warped product manifold introduced by Christianson and Wunsch, we encounter the first explicit example of a situation where an estimate with an algebraic loss of regularity exists and this loss is sharp. Due to the global-in-time nature of the estimate for the wave equation, the situation is more complicated than for the Schr\"{o}dinger equation. An initial estimate with sub-optimal loss is first obtained, where extra care is required due to the low frequency contributions. An improved estimate is then established using energy functionals that are inspired by WKB analysis. Finally, it is shown that the loss cannot be improved by any power by saturating the estimate with a quasimode.Comment: 18 page

An evaluation of kurtosis beamforming in magnetoencephalography to localize the epileptogenic zone in drug resistant epilepsy patients

OBJECTIVE: Kurtosis beamforming is a useful technique for analysing magnetoencephalograpy (MEG) data containing epileptic spikes. However, the implementation varies and few studies measure concordance with subsequently resected areas. We evaluated kurtosis beamforming as a means of localizing spikes in drug-resistant epilepsy patients. METHODS: We retrospectively applied kurtosis beamforming to MEG recordings of 22 epilepsy patients that had previously been analysed using equivalent current dipole (ECD) fitting. Virtual electrodes were placed in the kurtosis volumetric peaks and visually inspected to select a candidate source. The candidate sources were compared to the ECD localizations and resection areas. RESULTS: The kurtosis beamformer produced interpretable localizations in 18/22 patients, of which the candidate source coincided with the resection lobe in 9/13 seizure-free patients and in 3/5 patients with persistent seizures. The sublobar accuracy of the kurtosis beamformer with respect to the resection zone was higher than ECD (56% and 50%, respectively), however, ECD resulted in a higher lobar accuracy (75%, 67%). CONCLUSIONS: Kurtosis beamforming may provide additional value when spikes are not clearly discernible on the sensors and support ECD localizations when dipoles are scattered. SIGNIFICANCE: Kurtosis beamforming should be integrated with existing clinical protocols to assist in localizing the epileptogenic zone

The role of epidemic spreading in seizure dynamics and epilepsy surgery

AbstractEpilepsy surgery is the treatment of choice for drug-resistant epilepsy patients, but only leads to seizure freedom for roughly two in three patients. To address this problem, we designed a patient-specific epilepsy surgery model combining large-scale magnetoencephalography (MEG) brain networks with an epidemic spreading model. This simple model was enough to reproduce the stereo-tactical electroencephalography (SEEG) seizure propagation patterns of all patients (N = 15), when considering the resection areas (RA) as the epidemic seed. Moreover, the goodness of fit of the model predicted surgical outcome. Once adapted for each patient, the model can generate alternative hypothesis of the seizure onset zone and test different resection strategies in silico. Overall, our findings indicate that spreading models based on patient-specific MEG connectivity can be used to predict surgical outcomes, with better fit results and greater reduction on seizure propagation linked to higher likelihood of seizure freedom after surgery. Finally, we introduced a population model that can be individualized by considering only the patient-specific MEG network, and showed that it not only conserves but improves the group classification. Thus, it may pave the way to generalize this framework to patients without SEEG recordings, reduce the risk of overfitting and improve the stability of the analyses

Localization of the Epileptogenic Zone Using Interictal MEG and Machine Learning in a Large Cohort of Drug-Resistant Epilepsy Patients

Objective: Epilepsy surgery results in seizure freedom in the majority of drug-resistant patients. To improve surgery outcome we studied whether MEG metrics combined with machine learning can improve localization of the epileptogenic zone, thereby enhancing the chance of seizure freedom.Methods: Presurgical interictal MEG recordings of 94 patients (64 seizure-free >1y post-surgery) were analyzed to extract four metrics in source space: delta power, low-to-high-frequency power ratio, functional connectivity (phase lag index), and minimum spanning tree betweenness centrality. At the group level, we estimated the overlap of the resection area with the five highest values for each metric and determined whether this overlap differed between surgery outcomes. At the individual level, those metrics were used in machine learning classifiers (linear support vector machine (SVM) and random forest) to distinguish between resection and non-resection areas and between surgery outcome groups.Results: The highest values, for all metrics, overlapped with the resection area in more than half of the patients, but the overlap did not differ between surgery outcome groups. The classifiers distinguished the resection areas from non-resection areas with 59.94% accuracy (95% confidence interval: 59.67–60.22%) for SVM and 60.34% (59.98–60.71%) for random forest, but could not differentiate seizure-free from not seizure-free patients [43.77% accuracy (42.08–45.45%) for SVM and 49.03% (47.25–50.82%) for random forest].Significance: All four metrics localized the resection area but did not distinguish between surgery outcome groups, demonstrating that metrics derived from interictal MEG correspond to expert consensus based on several presurgical evaluation modalities, but do not yet localize the epileptogenic zone. Metrics should be improved such that they correspond to the resection area in seizure-free patients but not in patients with persistent seizures. It is important to test such localization strategies at an individual level, for example by using machine learning or individualized models, since surgery is individually tailored

MEG Beamformer-Based Reconstructions of Functional Networks in Mild Cognitive Impairment

Subjects with mild cognitive impairment (MCI) have an increased risk of developing Alzheimer’s disease (AD), and their functional brain networks are presumably already altered. To test this hypothesis, we compared magnetoencephalography (MEG) eyes-closed resting-state recordings from 29 MCI subjects and 29 healthy elderly subjects in the present exploratory study. Functional connectivity in different frequency bands was assessed with the phase lag index (PLI) in source space. Normalized weighted clustering coefficient (normalized Cw) and path length (normalized Lw), as well as network measures derived from the minimum spanning tree [MST; i.e., betweenness centrality (BC) and node degree], were calculated. First, we found altered PLI values in the lower and upper alpha bands in MCI patients compared to controls. Thereafter, we explored network differences in these frequency bands. Normalized Cw and Lw did not differ between the groups, whereas BC and node degree of the MST differed, although these differences did not survive correction for multiple testing using the False Discovery Rate (FDR). As an exploratory study, we may conclude that: (1) the increases and decreases observed in PLI values in lower and upper alpha bands in MCI patients may be interpreted as a dual pattern of disconnection and aberrant functioning; (2) network measures are in line with connectivity findings, indicating a lower efficiency of the brain networks in MCI patients; (3) the MST centrality measures are more sensitive to detect subtle differences in the functional brain networks in MCI than traditional graph theoretical metrics

Altering neuronal excitability to preserve network connectivity in a computational model of Alzheimer's disease

<div><p>Neuronal hyperactivity and hyperexcitability of the cerebral cortex and hippocampal region is an increasingly observed phenomenon in preclinical Alzheimer’s disease (AD). In later stages, oscillatory slowing and loss of functional connectivity are ubiquitous. Recent evidence suggests that neuronal dynamics have a prominent role in AD pathophysiology, making it a potentially interesting therapeutic target. However, although neuronal activity can be manipulated by various (non-)pharmacological means, intervening in a highly integrated system that depends on complex dynamics can produce counterintuitive and adverse effects. Computational dynamic network modeling may serve as a virtual test ground for developing effective interventions. To explore this approach, a previously introduced large-scale neural mass network with human brain topology was used to simulate the temporal evolution of AD-like, activity-dependent network degeneration. In addition, six defense strategies that either enhanced or diminished neuronal excitability were tested against the degeneration process, targeting excitatory and inhibitory neurons combined or separately. Outcome measures described oscillatory, connectivity and topological features of the damaged networks. Over time, the various interventions produced diverse large-scale network effects. Contrary to our hypothesis, the most successful strategy was a selective stimulation of all excitatory neurons in the network; it substantially prolonged the preservation of network integrity. The results of this study imply that functional network damage due to pathological neuronal activity can be opposed by targeted adjustment of neuronal excitability levels. The present approach may help to explore therapeutic effects aimed at preserving or restoring neuronal network integrity and contribute to better-informed intervention choices in future clinical trials in AD.</p></div

Slowing of Hippocampal Activity Correlates with Cognitive Decline in Early Onset Alzheimer's Disease. An MEG Study with Virtual Electrodes

Pathology in Alzheimer’s disease (AD) starts in the entorhinal cortex and hippocampus. Because of their deep location, activity from these areas is difficult to record with conventional electro- or magnetoencephalography (EEG/MEG). The purpose of this study was to explore hippocampal activity in AD patients and healthy controls using “virtual MEG electrodes”. We used resting-state MEG recordings from 27 early onset AD patients [age 60.6 ± 5.4, 12 females, mini-mental state examination (MMSE) range: 19–28] and 26 cognitively healthy age- and gender-matched controls (age 61.8 ± 5.5, 14 females). Activity was reconstructed using beamformer-based virtual electrodes for 78 cortical regions and 6 hippocampal regions. Group differences in peak frequency and relative power in six frequency bands were identified using permutation testing. For the patients, spearman correlations between the MMSE scores and peak frequency or relative power were calculated. Moreover, receiver operator characteristic curves were plotted to estimate the diagnostic accuracy. We found a lower hippocampal peak frequency in AD compared to controls, which, in the patients, correlated positively with MMSE [r(25) = 0.61; p < 0.01] whereas hippocampal relative theta power correlated negatively with MMSE [r(25) = -0.54; p < 0.01]. Cortical peak frequency was also lower in AD in association areas. Furthermore, cortical peak frequency correlated positively with MMSE [r(25) = 0.43; p < 0.05]. In line with this finding, relative theta power was higher in AD across the cortex, and relative alpha and beta power was lower in more circumscribed areas. The average cortical relative theta power was the best discriminator between AD and controls (sensitivity 82%; specificity 81%). Using beamformer-based virtual electrodes, we were able to detect hippocampal activity in AD. In AD, this hippocampal activity is slowed, and correlates better with cognition than the (slowed) activity in cortical areas. On the other hand, the average cortical relative power in the theta band was shown to be the best diagnostic discriminator. We postulate that this novel approach using virtual electrodes can be used in future research to quantify functional interactions between the hippocampi and cortical areas

Effect of the ‘stimulation of excitatory neurons’ intervention on structural connectivity.

<p>At three different time points, structural connectivity strength is compared with the ‘no intervention’ condition (blue lines). This is done in six different categories, based on structural connectivity level (degree). Highly connected hubs fall in the sixth category, most right in the chart. Note: error bars indicate standard deviations based on all node degrees per category. For a detailed list of region names and degree distributions see [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005707#pcbi.1005707.ref018" target="_blank">18</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005707#pcbi.1005707.t001" target="_blank">Table 1</a>].</p

Network topology.

<p>A) Network analysis: comparison of normalized clustering coefficient (gamma) values over time for the different interventions, indicating local connectivity strength. B) Comparison of modularity (community presence) over time for the different interventions. C) Comparison of algebraic connectivity values over time for the different interventions, indicating network robustness. D) Comparison of MST leaf number values over time for the different interventions, reflecting network hub presence. Error bars indicate the standard deviation over the total number of runs of the various strategies.</p

Overview of network analysis measures.

<p>Measure selection was based on the aim to describe the most robust and relevant functional network changes in AD. For exact definitions please refer to the method section and Supporting Information S4.</p