Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type

Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization

Electrophysiological characterization of the hyperdirect pathway and its functional relevance for subthalamic deep brain stimulation

The subthalamic nucleus (STN) receives input from various cortical areas via hyperdirect pathway (HDP) which bypasses the basal-ganglia loop. Recently, the HDP has gained increasing interest, because of its relevance for STN deep brain stimulation (DBS). To understand the HDP's role cortical responses evoked by STN-DBS have been investigated. These responses have short (<2 ms), medium (2–15 ms), and long (20–70 ms) latencies. Medium-latency responses are supposed to represent antidromic cortical activations via HDP. Together with long-latency responses the medium responses can potentially be used as biomarker of DBS efficacy as well as side effects. We here propose that the activation sequence of the cortical evoked responses can be conceptualized as high frequency oscillations (HFO) for signal analysis. HFO might therefore serve as marker for antidromic activation. Using existing knowledge on HFO recordings, this approach allows data analyses and physiological modeling to advance the pathophysiological understanding of cortical DBS-evoked high-frequency activity

Separating neural oscillations from aperiodic 1/f activity: Challenges and recommendations

Electrophysiological power spectra typically consist of two components: An aperiodic part usually following an 1/f power law [Formula: see text] and periodic components appearing as spectral peaks. While the investigation of the periodic parts, commonly referred to as neural oscillations, has received considerable attention, the study of the aperiodic part has only recently gained more interest. The periodic part is usually quantified by center frequencies, powers, and bandwidths, while the aperiodic part is parameterized by the y-intercept and the 1/f exponent [Formula: see text]. For investigation of either part, however, it is essential to separate the two components. In this article, we scrutinize two frequently used methods, FOOOF (Fitting Oscillations & One-Over-F) and IRASA (Irregular Resampling Auto-Spectral Analysis), that are commonly used to separate the periodic from the aperiodic component. We evaluate these methods using diverse spectra obtained with electroencephalography (EEG), magnetoencephalography (MEG), and local field potential (LFP) recordings relating to three independent research datasets. Each method and each dataset poses distinct challenges for the extraction of both spectral parts. The specific spectral features hindering the periodic and aperiodic separation are highlighted by simulations of power spectra emphasizing these features. Through comparison with the simulation parameters defined a priori, the parameterization error of each method is quantified. Based on the real and simulated power spectra, we evaluate the advantages of both methods, discuss common challenges, note which spectral features impede the separation, assess the computational costs, and propose recommendations on how to use them

Bifurcation of critical points along gap-continuous families of subspaces

We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families, and apply our results to semilinear systems of ordinary differential equations

Power-law dynamics in cortical excitability as probed by early somatosensory evoked responses

While it is well-established that instantaneous changes in neuronal networks’ states lead to variability in brain responses and behavior, the mechanisms causing this variability are poorly understood. Insights into the organization of underlying system dynamics may be gained by examining the temporal structure of network state fluctuations, such as reflected in instantaneous cortical excitability. Using the early part of single-trial somatosensory evoked potentials in the human EEG, we non-invasively tracked the magnitude of excitatory post-synaptic potentials in the primary somatosensory cortex (BA 3b) in response to median nerve stimulation. Fluctuations in cortical excitability demonstrated long-range temporal dependencies decaying according to a power-law across trials. As these dynamics covaried with pre-stimulus alpha oscillations, we establish a functional link between ongoing and evoked activity and argue that the co-emergence of similar temporal power-laws may originate from neuronal networks poised close to a critical state, representing a parsimonious organizing principle of neural variability

The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems

We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author

A K-theoretical Invariant and Bifurcation for Homoclinics of Hamiltonian Systems

We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to apply this invariant to investigate bifurcation of homoclinic solutions of families of Hamiltonian systems which are parametrised by tori

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.Comment: 34 pages, 13 figures, 45 references, to appear in Ann Glob Anal Geom. The final publication will be available at http://www.springerlink.com. arXiv admin note: substantial text overlap with arXiv:math/040613

Mixed-mode liquid chromatography for the rapid analysis of biocatalytic glucaric acid reaction pathways.

Glucaric acid (GlucA) has been identified as one of the top 10 potential bio-based chemicals for replacement of oil-based chemicals. Several synthetic enzyme pathways have been engineered in bacteria and yeast to produce GlucA from glucose and myo-inositol. However, the yields and titres achieved with these systems remain too low for the requirements of a bio-based GlucA industry. A major limitation for the optimisation of GlucA production via synthetic enzymatic pathways are the laborious analytical procedures required to detect the final product (GlucA) and pathway intermediates. We have developed a novel method for the simple and simultaneous analysis of GlucA and pathway intermediates to address this limitation using mixed mode (MM) HILIC and weak anion exchange chromatography (WAX), referred to as MM HILIC/WAX, coupled with RID. Isocratic mobile phase conditions and the sample solvent were optimised for the separation of GlucA, glucose-1-phosphate (G1P), glucose-6-phosphate (G6P), inositol-1-phosphate (I1P), myo-inositol and glucuronic acid (GA). The method showed good repeatability, precision and excellent accuracy with detection and quantitation limits (LOD and LOQ) of 1.5-2 and 577 mM, respectively. The method developed was used for monitoring the enzymatic synthesis of the final step in the GlucA pathway, and showed that GlucA was produced from GA with near 100% conversion and a titre of 9.2 g L-1