Multimodal oscillations in systems with strong contraction

One- and two-parameter families of flows in $R^3$ near an Andronov-Hopf bifurcation (AHB) are investigated in this work. We identify conditions on the global vector field, which yield a rich family of multimodal orbits passing close to a weakly unstable saddle-focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle-focus. Our analysis covers both cases of sub- and supercritical AHB. For the supercritical case, we find that the periodic orbits born from the AHB are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium. If the AHB is subcritical, it is accompanied by the appearance of multimodal orbits, which consist of long series of nearly harmonic oscillations separated by large amplitude spikes. We analyze the dependence of the interspike intervals (which can be extremely long) on the control parameters. In particular, we show that the interspike intervals grow logarithmically as the boundary between regions of sub- and supercritical AHB is approached in the parameter space. We also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub- and supercritical AHB and explain the mechanism generating these oscillations. This work is motivated by the numerical results for a finite-dimensional approximation of a free boundary problem modeling solid fuel combustion

Excitable Dynamics in the Presence of Time Delay

The spiking properties of a subcritical Hopf oscillator with a time delayed nonlinear feedback is investigated. Finite time delay is found to significantly affect both the statistics and the fine structure of the spiking behavior. These dynamical changes are explained in terms of the fundamental modifications occurring in the bifurcation scenario of the system. Our mathematical model can find useful applications in understanding the dynamical behaviour of various real life excitable systems where propagation delay effects are ubiquitous.Comment: 10 pages including 7 figure

Tensor Analysis and Fusion of Multimodal Brain Images

Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks that underpin human cognitive processes by analysis of different neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the multimodal, multi-scale nature of neuroimaging data is well reflected by a multi-way (tensor) structure where the underlying processes can be summarized by a relatively small number of components or "atoms". We introduce Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network notation in order to analyze these models. These diagrams not only clarify matrix and tensor EEG and fMRI time/frequency analysis and inverse problems, but also help understand multimodal fusion via Multiway Partial Least Squares and Coupled Matrix-Tensor Factorization. We show here, for the first time, that Granger causal analysis of brain networks is a tensor regression problem, thus allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI recordings shows the potential of the methods and suggests their use in other scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE

Optimal set of EEG features for emotional state classification and trajectory visualization in Parkinson's disease

In addition to classic motor signs and symptoms, individuals with Parkinson's disease (PD) are characterized by emotional deficits. Ongoing brain activity can be recorded by electroencephalograph (EEG) to discover the links between emotional states and brain activity. This study utilized machine-learning algorithms to categorize emotional states in PD patients compared with healthy controls (HC) using EEG. Twenty non-demented PD patients and 20 healthy age-, gender-, and education level-matched controls viewed happiness, sadness, fear, anger, surprise, and disgust emotional stimuli while fourteen-channel EEG was being recorded. Multimodal stimulus (combination of audio and visual) was used to evoke the emotions. To classify the EEG-based emotional states and visualize the changes of emotional states over time, this paper compares four kinds of EEG features for emotional state classification and proposes an approach to track the trajectory of emotion changes with manifold learning. From the experimental results using our EEG data set, we found that (a) bispectrum feature is superior to other three kinds of features, namely power spectrum, wavelet packet and nonlinear dynamical analysis; (b) higher frequency bands (alpha, beta and gamma) play a more important role in emotion activities than lower frequency bands (delta and theta) in both groups and; (c) the trajectory of emotion changes can be visualized by reducing subject-independent features with manifold learning. This provides a promising way of implementing visualization of patient's emotional state in real time and leads to a practical system for noninvasive assessment of the emotional impairments associated with neurological disorders

The Hunt for Exomoons with Kepler (HEK): I. Description of a New Observational Project

Two decades ago, empirical evidence concerning the existence and frequency of planets around stars, other than our own, was absent. Since this time, the detection of extrasolar planets from Jupiter-sized to most recently Earth-sized worlds has blossomed and we are finally able to shed light on the plurality of Earth-like, habitable planets in the cosmos. Extrasolar moons may also be frequent habitable worlds but their detection or even systematic pursuit remains lacking in the current literature. Here, we present a description of the first systematic search for extrasolar moons as part of a new observational project called "The Hunt for Exomoons with Kepler" (HEK). The HEK project distills the entire list of known transiting planet candidates found by Kepler (2326 at the time of writing) down to the most promising candidates for hosting a moon. Selected targets are fitted using a multimodal nested sampling algorithm coupled with a planet-with-moon light curve modelling routine. By comparing the Bayesian evidence of a planet-only model to that of a planet-with-moon, the detection process is handled in a Bayesian framework. In the case of null detections, upper limits derived from posteriors marginalised over the entire prior volume will be provided to inform the frequency of large moons around viable planetary hosts, eta-moon. After discussing our methodologies for target selection, modelling, fitting and vetting, we provide two example analyses.Comment: 21 pages, 8 figures, 4 tables, accepted in Ap