The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in \RR^2 through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two $C^0$ normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between $C^0$ tori located at the same energy levels. By means of the {\em impact map} we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the {\em scattering map}, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnol'd diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring

The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter $\epsilon$ uncouples the system at $\epsilon=0$. Using a normal form for $N=2$ identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson-Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in existing models). Using numerical continuation we confirm our theoretical analysis for small coupling strength and explore the bifurcation diagrams for large coupling strength, where the normal form approximation breaks down

Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon "beyond all orders". In this paper we provide a formula for the distance between the corresponding stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings

Cholesterol Induces Specific Spatial and Orientational Order in Cholesterol/Phospholipid Membranes

In lipid bilayers, cholesterol facilitates the formation of the liquid-ordered phase and enables the formation of laterally ordered structures such as lipid rafts. While these domains have an important role in a variety of cellular processes, the precise atomic-level mechanisms responsible for cholesterol's specific ordering and packing capability have remained unresolved

Cholesterol Induces Specific Spatial and Orientational Order in Cholesterol/Phospholipid Membranes

In lipid bilayers, cholesterol facilitates the formation of the liquid-ordered phase and enables the formation of laterally ordered structures such as lipid rafts. While these domains have an important role in a variety of cellular processes, the precise atomic-level mechanisms responsible for cholesterol's specific ordering and packing capability have remained unresolved

Direct Integration and Non-Perturbative Effects in Matrix Models

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.Comment: 51 pages, 8 figure

Neural responses to others’ pain vary with psychopathic traits in healthy adult males

Disrupted empathic processing is a core feature of psychopathy. Neuroimaging data have suggested that individuals with high levels of psychopathic traits show atypical responses to others' pain in a network of brain regions typically recruited during empathic processing (anterior insula, inferior frontal gyrus, and mid- and anterior cingulate cortex). Here, we investigated whether neural responses to others' pain vary with psychopathic traits within the general population in a similar manner to that found in individuals at the extreme end of the continuum. As predicted, variation in psychopathic traits was associated with variation in neural responses to others' pain in the network of brain regions typically engaged during empathic processing. Consistent with previous research, our findings indicated the presence of suppressor effects in the association of levels of the affective-interpersonal and lifestyle-antisocial dimensions of psychopathy with neural responses to others' pain. That is, after controlling for the influence of the other dimension, higher affective-interpersonal psychopathic traits were associated with reduced neural responses to others' pain, whilst higher lifestyle-antisocial psychopathic traits were associated with increased neural responses to others' pain. Our findings provide further evidence that atypical function in this network might represent neural markers of disrupted emotional and empathic processing; that the two dimensions of psychopathy might tap into distinct underlying vulnerabilities; and, most importantly, that the relationships observed at the extreme end of the psychopathy spectrum apply to the nonclinical distribution of these traits, providing further evidence for continuities in the mechanisms underlying psychopathic traits across the general population

Advanced MR techniques for preoperative glioma characterization: Part 1

Preoperative clinical magnetic resonance imaging (MRI) protocols for gliomas, brain tumors with dismal outcomes due to their infiltrative properties, still rely on conventional structural MRI, which does not deliver information on tumor genotype and is limited in the delineation of diffuse gliomas. The GliMR COST action wants to raise awareness about the state of the art of advanced MRI techniques in gliomas and their possible clinical translation or lack thereof. This review describes current methods, limits, and applications of advanced MRI for the preoperative assessment of glioma, summarizing the level of clinical validation of different techniques. In this first part, we discuss dynamic susceptibility contrast and dynamic contrast-enhanced MRI, arterial spin labeling, diffusion-weighted MRI, vessel imaging, and magnetic resonance fingerprinting. The second part of this review addresses magnetic resonance spectroscopy, chemical exchange saturation transfer, susceptibility-weighted imaging, MRI-PET, MR elastography, and MR-based radiomics applications. Evidence Level: 3 Technical Efficacy: Stage 2