Frequency-based brain networks: From a multiplex framework to a full multilayer description

We explore how to study dynamical interactions between brain regions using functional multilayer networks whose layers represent the different frequency bands at which a brain operates. Specifically, we investigate the consequences of considering the brain as a multilayer network in which all brain regions can interact with each other at different frequency bands, instead of as a multiplex network, in which interactions between different frequency bands are only allowed within each brain region and not between them. We study the second smallest eigenvalue of the combinatorial supra-Laplacian matrix of the multilayer network in detail, and we thereby show that the heterogeneity of interlayer edges and, especially, the fraction of missing edges crucially modify the spectral properties of the multilayer network. We illustrate our results with both synthetic network models and real data sets obtained from resting state magnetoencephalography. Our work demonstrates an important issue in the construction of frequency-based multilayer brain networks.Comment: 13 pages, 8 figure

Developmentally regulated multisensory integration for prey localization in the medicinal leech

Medicinal leeches, like many aquatic animals, use water disturbances to localize their prey, so they need to be able to determine if a wave disturbance is created by prey or by another source. Many aquatic predators perform this separation by responding only to those wave frequencies representing their prey. As leeches' prey preference changes over the course of their development, we examined their responses at three different life stages. We found that juveniles more readily localize wave sources of lower frequencies (2 Hz) than their adult counterparts (8–12 Hz), and that adolescents exhibited elements of both juvenile and adult behavior, readily localizing sources of both frequencies. Leeches are known to be able to localize the source of waves through the use of either mechanical or visual information. We separately characterized their ability to localize various frequencies of stimuli using unimodal cues. Within a single modality, the frequency–response curves of adults and juveniles were virtually indistinguishable. However, the differences between the responses for each modality (visual and mechanosensory) were striking. The optimal visual stimulus had a much lower frequency (2 Hz) than the optimal mechanical stimulus (12 Hz). These frequencies matched, respectively, the juvenile and the adult preferred frequency for multimodally sensed waves. This suggests that, in the multimodal condition, adult behavior is driven more by mechanosensory information and juvenile behavior more by visual. Indeed, when stimuli of the two modalities were placed in conflict with one another, adult leeches, unlike juveniles, were attracted to the mechanical stimulus much more strongly than to the visual stimulus

PT-symmetry broken by point-group symmetry

We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schr\"{o}dinger equation for a particle in a square box with the PT-symmetric potential $V(x,y)=iaxy$. Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of $|a|$. Point-group symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schr\"odinger equation with the potential $V(x,y)=iaxy^{2}$ exhibits real eigenvalues for sufficiently small values of $|a|$. Point group symmetry suggests that PT-symmetry may be broken in the former case and unbroken in the latter one

A uniqueness criterion for the Fock quantization of scalar fields with time dependent mass

A major problem in the quantization of fields in curved spacetimes is the ambiguity in the choice of a Fock representation for the canonical commutation relations. There exists an infinite number of choices leading to different physical predictions. In stationary scenarios, a common strategy is to select a vacuum (or a family of unitarily equivalent vacua) by requiring invariance under the spacetime symmetries. When stationarity is lost, a natural generalization consists in replacing time invariance by unitarity in the evolution. We prove that, when the spatial sections are compact, the criterion of a unitary dynamics, together with the invariance under the spatial isometries, suffices to select a unique family of Fock quantizations for a scalar field with time dependent mass.Comment: 11 pages, version accepted for publication in Classical and Quantum Gravit

Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes

For Klein-Gordon fields, it is well known that there exist an infinite number of nonequivalent Fock representations of the canonical commutation relations and, therefore, of inequivalent quantum theories. A context in which this kind of ambiguities arises and prevents the derivation of robust results is, e.g., in the quantum analysis of cosmological perturbations. In these situations, typically, a suitable scaling of the field by a time dependent function leads to a description in an auxiliary static background, though the nonstationarity still shows up in a time dependent mass. For such a field description, and assuming the compactness of the spatial sections, we recently proved in three or less spatial dimensions that the criteria of a natural implementation of the spatial symmetries and of a unitary time evolution are able to select a unique class of unitarily equivalent vacua, and hence of Fock representations. In this work, we succeed to extend our uniqueness result to the consideration of all possible field descriptions that can be reached by a time dependent canonical transformation which, in particular, involves a scaling of the field by a function of time. This kind of canonical transformations modify the dynamics of the system and introduce a further ambiguity in its quantum description, exceeding the choice of a Fock representation. Remarkably, for any compact spatial manifold in less than four dimensions, we show that our criteria eliminate any possible nontrivial scaling of the field other than that leading to the description in an auxiliary static background. Besides, we show that either no time dependent redefinition of the field momentum is allowed or, if this may happen, the redefinition does not introduce any Fock representation that cannot be obtained by a unitary transformation.Comment: 37 pages. Modified title. Improved discussion concerning the spatial symmetry group. New section (section VI

Memory effects in vibrated granular systems

Granular materials present memory effects when submitted to tapping processes. These effects have been observed experimentally and are discussed here in the context of a general kind of model systems for compaction formulated at a mesoscopic level. The theoretical predictions qualitatively agree with the experimental results. As an example, a particular simple model is used for detailed calculations.Comment: 12 pages, 5 figures; to appear in Journal of Physics: Condensed Matter (Special Issue: Proceedings of ESF SPHINX Workshop on ``Glassy behaviour of kinetically constrained models.''

Mesoscopic Theory of Critical Fluctuations in Isolated Granular Gases

Fluctuating hydrodynamics is used to describe the total energy fluctuations of a freely evolving gas of inelastic hard spheres near the threshold of the clustering instability. They are shown to be governed by vorticity fluctuations only, that also lead to a renormalization of the average total energy. The theory predicts a power-law divergent behavior of the scaled second moment of the fluctuations, and a scaling property of their probability distribution, both in agreement with simulations results. A more quantitative comparison between theory and simulation for the critical amplitudes and the form of the scaling function is also carried out