Well-Posedness and Symmetries of Strongly Coupled Network Equations

We consider a diffusion process on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schroedinger equations on a quantum graph are discussed.Comment: 25 pages. Corrected typos and minor change

Multiple dynamical time-scales in networks with hierarchically nested modular organization

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intra-modular and slow inter-modular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.Comment: 10 pages, 4 figure

Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions

In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Mat\'ern functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.Comment: 26 page

Synchronization, Diversity, and Topology of Networks of Integrate and Fire Oscillators

We study synchronization dynamics of a population of pulse-coupled oscillators. In particular, we focus our attention in the interplay between networks topological disorder and its synchronization features. Firstly, we analyze synchronization time $T$ in random networks, and find a scaling law which relates $T$ to networks connectivity. Then, we carry on comparing synchronization time for several other topological configurations, characterized by a different degree of randomness. The analysis shows that regular lattices perform better than any other disordered network. The fact can be understood by considering the variability in the number of links between two adjacent neighbors. This phenomenon is equivalent to have a non-random topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings.Comment: 6 pages, 8 figures, LaTeX 209, uses RevTe

Topological Photonics

Topology is revolutionizing photonics, bringing with it new theoretical discoveries and a wealth of potential applications. This field was inspired by the discovery of topological insulators, in which interfacial electrons transport without dissipation even in the presence of impurities. Similarly, new optical mirrors of different wave-vector space topologies have been constructed to support new states of light propagating at their interfaces. These novel waveguides allow light to flow around large imperfections without back-reflection. The present review explains the underlying principles and highlights the major findings in photonic crystals, coupled resonators, metamaterials and quasicrystals.Comment: progress and review of an emerging field, 12 pages, 6 figures and 1 tabl

Bistable Percepts in the Brain: fMRI Contrasts Monocular Pattern Rivalry and Binocular Rivalry

The neural correlates of binocular rivalry have been actively debated in recent years, and are of considerable interest as they may shed light on mechanisms of conscious awareness. In a related phenomenon, monocular rivalry, a composite image is shown to both eyes. The subject experiences perceptual alternations in which the two stimulus components alternate in clarity or salience. The experience is similar to perceptual alternations in binocular rivalry, although the reduction in visibility of the suppressed component is greater for binocular rivalry, especially at higher stimulus contrasts. We used fMRI at 3T to image activity in visual cortex while subjects perceived either monocular or binocular rivalry, or a matched non-rivalrous control condition. The stimulus patterns were left/right oblique gratings with the luminance contrast set at 9%, 18% or 36%. Compared to a blank screen, both binocular and monocular rivalry showed a U-shaped function of activation as a function of stimulus contrast, i.e. higher activity for most areas at 9% and 36%. The sites of cortical activation for monocular rivalry included occipital pole (V1, V2, V3), ventral temporal, and superior parietal cortex. The additional areas for binocular rivalry included lateral occipital regions, as well as inferior parietal cortex close to the temporoparietal junction (TPJ). In particular, higher-tier areas MT+ and V3A were more active for binocular than monocular rivalry for all contrasts. In comparison, activation in V2 and V3 was reduced for binocular compared to monocular rivalry at the higher contrasts that evoked stronger binocular perceptual suppression, indicating that the effects of suppression are not limited to interocular suppression in V1

Adiabatic theorems for generators of contracting evolutions

We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.Comment: 31 pages, 4 figures, replaced by the version accepted for publication in CM

Activity in Face-Responsive Brain Regions is Modulated by Invisible, Attended Faces: Evidence from Masked Priming

It is often assumed that neural activity in face-responsive regions of primate cortex correlates with conscious perception of faces. However, whether such activity occurs without awareness is still debated. Using functional magnetic resonance imaging (fMRI) in conjunction with a novel masked face priming paradigm, we observed neural modulations that could not be attributed to perceptual awareness. More specifically, we found reduced activity in several classic face-processing regions, including the “fusiform face area,” “occipital face area,” and superior temporal sulcus, when a face was preceded by a briefly flashed image of the same face, relative to a different face, even when 2 images of the same face differed. Importantly, unlike most previous studies, which have minimized awareness by using conditions of inattention, the present results occurred when the stimuli (the primes) were attended. By contrast, when primes were perceived consciously, in a long-lag priming paradigm, we found repetition-related activity increases in additional frontal and parietal regions. These data not only demonstrate that fMRI activity in face-responsive regions can be modulated independently of perceptual awareness, but also document where such subliminal face-processing occurs (i.e., restricted to face-responsive regions of occipital and temporal cortex) and to what extent (i.e., independent of the specific image)

The Temporal Winner-Take-All Readout

How can the central nervous system make accurate decisions about external stimuli at short times on the basis of the noisy responses of nerve cell populations? It has been suggested that spike time latency is the source of fast decisions. Here, we propose a simple and fast readout mechanism, the temporal Winner-Take-All (tWTA), and undertake a study of its accuracy. The tWTA is studied in the framework of a statistical model for the dynamic response of a nerve cell population to an external stimulus. Each cell is characterized by a preferred stimulus, a unique value of the external stimulus for which it responds fastest. The tWTA estimate for the stimulus is the preferred stimulus of the cell that fired the first spike in the entire population. We then pose the questions: How accurate is the tWTA readout? What are the parameters that govern this accuracy? What are the effects of noise correlations and baseline firing? We find that tWTA sensitivity to the stimulus grows algebraically fast with the number of cells in the population, N, in contrast to the logarithmic slow scaling of the conventional rate-WTA sensitivity with N. Noise correlations in first-spike times of different cells can limit the accuracy of the tWTA readout, even in the limit of large N, similar to the effect that has been observed in population coding theory. We show that baseline firing also has a detrimental effect on tWTA accuracy. We suggest a generalization of the tWTA, the n-tWTA, which estimates the stimulus by the identity of the group of cells firing the first n spikes and show how this simple generalization can overcome the detrimental effect of baseline firing. Thus, the tWTA can provide fast and accurate responses discriminating between a small number of alternatives. High accuracy in estimation of a continuous stimulus can be obtained using the n-tWTA

The Spatial Origin of a Perceptual Transition in Binocular Rivalry

When the left and the right eye are simultaneously presented with incompatible images at overlapping retinal locations, an observer typically reports perceiving only one of the two images at a time. This phenomenon is called binocular rivalry. Perception during binocular rivalry is not stable; one of the images is perceptually dominant for a certain duration (typically in the order of a few seconds) after which perception switches towards the other image. This alternation between perceptual dominance and suppression will continue for as long the images are presented. A characteristic of binocular rivalry is that a perceptual transition from one image to the other generally occurs in a gradual manner: the image that was temporarily suppressed will regain perceptual dominance at isolated locations within the perceived image, after which its visibility spreads throughout the whole image. These gradual transitions from perceptual suppression to perceptual dominance have been labeled as traveling waves of perceptual dominance. In this study we investigate whether stimulus parameters affect the location at which a traveling wave starts. We varied the contrast, spatial frequency or motion speed in one of the rivaling images, while keeping the same parameter constant in the other image. We used a flash-suppression paradigm to force one of the rival images into perceptual suppression. Observers waited until the suppressed image became perceptually dominant again, and indicated the position at which this breakthrough from suppression occurred. Our results show that the starting point of a traveling wave during binocular rivalry is highly dependent on local stimulus parameters. More specifically, a traveling wave most likely started at the location where the contrast of the suppressed image was higher than that of the dominant one, the spatial frequency of the suppressed image was lower than that of the dominant one, and the motion speed of the suppressed image was higher than that of the dominant one. We suggest that a breakthrough from suppression to dominance occurs at the location where salience (the degree to which a stimulus element stands out relative to neighboring elements) of the suppressed image is higher than that of the dominant one. Our results further show that stimulus parameters affecting the temporal dynamics during continuous viewing of rival images described in other studies, also affect the spatial origin of traveling waves during binocular rivalry