Bimodal Maps Research Articles

Comparative fMRI and MEG localization of cortical sensorimotor function: Bimodal mapping supports motor area reorganization in glioma patients.

IntroductionPreoperative functional mapping in the vicinity of brain lesion is of high importance for avoiding complications in surgical management. However, space-occupying lesions may lead to functional reorganization or decreased BOLD activity.MethodsTherefore in 13 patients with cerebral gliomas or brain arterio-venous malformations/ hemangioma fMRI- and MEG-based cortical localizations of motor and somatosensory cortical activation pattern were compared in order to investigate their congruency.ResultsLocalization of cortical sensorimotor areas with fMRI and MEG showed good congruency with a mean spatial distance of around 10 mm, with differences depending on the localization method. The smallest mean differences for the centroids were found for MEF with MNE 8 mm and SEF with sLORETA 8 mm. Primary motor area (M1) reorganization was found in 5 of 12 patients in fMRI and confirmed with MEG data. In these 5 patients with M1-reorganization the distance between the border of the fMRI-based cortical M1-localization and the tumor border on T1w MR images varied between 0–4 mm, which was significant (P = 0.025) different to the distance in glioma patients without M1-reorganization (5–26 mm).ConclusionOur multimodal preoperative mapping approach combining fMRI and MEG reveals a high degree of spatial congruence and provided high evidence for the presence of motor cortex reorganization.

Animal movement: symbolic dynamics and topological classification.

We present a deterministic discrete dynamical system, which is used to produce and classify a variety of types of movements. The dynamical system is determined by the iteration of a bimodal interval map, dependent on one real parameter, up to scaling. The characterization of the movements is based on the topological classification of the discrete dynamical system. Techniques from symbolic dynamics and topological Markov chains are used. With this approach we obtain efficient simulation capability and a very simple characterization of distinct types of regular or complex motion. We discuss the application of the method to describe animal movement.

Non-monotone periodic orbits of a rotational horseshoe

In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair$(r,s)$of rational numbers with$r<s$in$(0,1)$, there exists a non-monotone orbit$B_{r,s}$in this family which has pseudo-Anosov type and rotation interval$[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits$H_{p/q}$, which are the largest for the forcing order among all the$(p,q)$-orbits of the rotational horseshoe.

Bifurcation Structures in a Bimodal Piecewise Linear Map

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

On Localized Vapor Pressure Gradients Governing Condensation and Frost Phenomena.

Interdroplet vapor pressure gradients are the driving mechanism for several phase-change phenomena such as condensation dry zones, interdroplet ice bridging, dry zones around ice, and frost halos. Despite the fundamental nature of the underlying pressure gradients, the majority of studies on these emerging phenomena have been primarily empirical. Using classical nucleation theory and Becker-Döring embryo formation kinetics, here we calculate the pressure field for all possible modes of condensation and desublimation in order to gain fundamental insight into how pressure gradients govern the behavior of dry zones, condensation frosting, and frost halos. Our findings reveal that in a variety of phase-change systems the thermodynamically favorable mode of nucleation can switch between condensation and desublimation depending upon the temperature and wettability of the surface. The calculated pressure field is used to model the length of a dry zone around liquid or ice droplets over a broad parameter space. The long-standing question of whether the vapor pressure at the interface of growing frost is saturated or supersaturated is resolved by considering the kinetics of interdroplet ice bridging. Finally, on the basis of theoretical calculations, we propose that there exists a new mode of frost halo that is yet to be experimentally observed; a bimodal phase map is developed, demonstrating its dependence on the temperature and wettability of the underlying substrate. We hope that the model and predictions contained herein will assist future efforts to exploit localized vapor pressure gradients for the design of spatially controlled or antifrosting phase-change systems.

Generalized Hertz model for bimodal nanomechanical mapping.

Bimodal atomic force microscopy uses a cantilever that is simultaneously driven at two of its eigenmodes (resonant modes). Parameters associated with both resonances can be measured and used to extract quantitative nanomechanical information about the sample surface. Driving the first eigenmode at a large amplitude and a higher eigenmode at a small amplitude simultaneously provides four independent observables that are sensitive to the tip–sample nanomechanical interaction parameters. To demonstrate this, a generalized theoretical framework for extracting nanomechanical sample properties from bimodal experiments is presented based on Hertzian contact mechanics. Three modes of operation for measuring cantilever parameters are considered: amplitude, phase, and frequency modulation. The experimental equivalence of all three modes is demonstrated on measurements of the second eigenmode parameters. The contact mechanics theory is then extended to power-law tip shape geometries, which is applied to analyze the experimental data and extract a shape and size of the tip interacting with a polystyrene surface.

Systematic Multidimensional Quantification of Nanoscale Systems From Bimodal Atomic Force Microscopy Data.

Here we explore the raw parameter space in air in bimodal atomic force microscopy (AFM) in order to enhance resolution, provide multiparameter maps, and produce suitable transformations that lead to physically intuitive maps general enough to be recognized by the broader community, i.e., stiffness, Hamaker constant, and adhesion force. We further consider model free transforms to enhance the raw parameter space in the form of alternative and more intelligible contrast maps. We employ highly oriented pyrolytic graphite, calcite, polypropylene, and dsDNA on mica to demonstrate a systematic form of parameter expansion. The proposed methodology to enhance and interpret a larger parameter space introduces a methodology to tractable multidimensional AFM from raw bimodal AFM maps.

Emotional head motion predicting from prosodic and linguistic features

Emotional head motion plays an important role in human-computer interaction (HCI), which is one of the important factors to improve users' experience in HCI. However, it is still not clear how head motions are influenced by speech features in different emotion states. In this study, we aim to construct a bimodal mapping model from speech to head motions, and try to discover what kinds of prosodic and linguistic features have the most significant influence on emotional head motions. A two-layer clustering schema is introduced to obtain reliable clusters from head motion parameters. With these clusters, an emotion related speech to head gesture mapping model is constructed by a Classification and Regression Tree (CART). Based on the statistic results of CART, a systematical statistic map of the relationship between speech features (including prosodic and linguistic features) and head gestures is presented. The map reveals the features which have the most significant influence on head motions in long or short utterances. We also make an analysis on how linguistic features contribute to different emotional expressions. The discussions in this work provide important references for realistic animation of speech driven talking-head or avatar.

Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

Discrete time price adjustment processes may fail to converge and may exhibit periodic or even chaotic behavior. To avoid large price changes, a version of the discrete time tâtonnement process for reaching an equilibrium in a pure exchange economy based on a cautious updating of the prices has been proposed two decades ago. This modification leads to a one dimensional bimodal piecewise smooth map, for which we show analytically that degenerate bifurcations and border collision bifurcations play a fundamental role for the asymptotic behavior of the model.

Variability in suspended sediment concentration in the Minas Basin, Bay of Fundy, and implications for changes due to tidal power extraction

The Bay of Fundy in eastern Canada has the world's largest tidal range of over 16m with tidal currents up to 5ms−1 making it an ideal place for tidal power extraction using tidal in-stream energy conversion devices in the Minas Passage. Field observations collected from ship-based and bottom-moored sensors over an 8-day period in 2013 are used to validate a 3D hydrodynamic and sediment transport model of the Minas Basin with measurements of water levels, current profiles, waves, suspended sediment fluxes and suspended sediment concentration (SSC) profiles. The sediment conditions are initialized using a bi-modal sediment distribution map and the model simulates both cohesive and non-cohesive sediments in the Minas Basin. Model results for fine-grained suspended sediment concentrations are compared horizontally, vertically, and temporally to observations and indicate strong data-model agreement for SSC from 5 to 287mgL−1. The implications of constructing a large-scale turbine farm within the Minas Passage and the impacts on suspended sediment within the Minas Basin are investigated using the model. The farm is simulated by adding semi-permeable structures that use an energy loss term in the fluid momentum equations to parameterize turbine regions in the hydrodynamic model. The results emphasize the sensitivity of the system to changes in flow and suggest that a large-scale tidal energy farm that extracts maximum power could reduce SSC by 37% on average across the basin which would influence physical and biological processes particularly on the fine-grained intertidal areas around the macrotidal basin.

Bandcount adding structure and collapse of chaotic attractors in a piecewise linear bimodal map

In this work we investigate bifurcation structures in the chaotic domain of a piecewise linear bimodal map. The map represents a model of a circuit proposed to generate chaotic signals. For practical purposes it is necessary that the map generates robust broad-band chaos. However, experiments show that this requirement is fulfilled not everywhere. We show that the chaotic domain in the parameter space of this map contains regions in which the map has multi-band chaotic attractors. These regions are confined by bifurcation curves associated with homoclinic bifurcations of unstable cycles, and form a so-called bandcount adding structure previously reported to occur in discontinuous maps. Additionally, it is shown that inside each of these regions chaotic attractors collapse to particular cycles existing on a domain of zero measure in the parameter space and organized in a period adding structure in the form known for circle maps.

Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

Tâtonnement processes are usually interpreted as auctions, where a fictitious agent sets the prices until an equilibrium is reached and the trades are made. The main purpose of such processes is to explain how an economy comes to its equilibrium. It is well known that discrete time price adjustment processes may fail to converge and may exhibit periodic or even chaotic behavior. To avoid large price changes, a version of the discrete time tâtonnement process for reaching an equilibrium in a pure exchange economy based on a cautious updating of the prices has been proposed two decades ago. This modification leads to a one dimensional bimodal piecewise smooth map, for which we show analytically that degenerate bifurcations and border collision bifurcations play a fundamental role for the asymptotic behavior of the model.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure). By crossing these boundaries the map displays robust chaos.

Harvest timing and its population dynamic consequences in a discrete single-species model

The timing of harvesting is a key instrument in managing and exploiting biological populations and renewable resources. Yet, there is little theory on harvest timing, and even less is known about the impact of different harvest times on the stability of population dynamics, even though this may drive population variability and risk of extinction. Here, we employ the framework proposed by Seno to study how harvesting at specific moments in the reproductive season affects not only population size but also stability. For populations with overcompensation, intermediate harvest times tend to be stabilizing (by simplifying dynamics in the case of unimodal maps and by preventing bubbling in the case of bimodal maps). For populations with a strong Allee effect, however, intermediate harvest times can have a twofold effect. On the one hand, they facilitate population persistence (if harvesting effort is low). On the other hand, they provoke population extinction (if harvesting effort is high). Early harvesting, currently considered common sense to take advantage of compensatory effects, may cut into the breeding stock when the population has not yet surpassed the critical Allee threshold. The results in this paper highlight, for the first time, the crucial interplay between harvest timing and Allee effects. Moreover, they demonstrate that harvesting with the same effort but at different moments in time can dramatically alter the impact on the population.

BIFURCATION STRUCTURES IN A BIMODAL PIECEWISE LINEAR MAP: REGULAR DYNAMICS

We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles having at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique.

UNIVERSAL FORM OF RENORMALIZABLE KNOTS IN SYMBOLIC DYNAMICS OF BIMODAL MAPS

A general relation between knot theory and symbolic dynamics is studied for bimodal maps in this paper. When symbolic sequences of maps are expressed as knots, it is easy to see that the knot of a renormalizable sequence is composed of a bunch of periodic flows. In this setting, the generation of renormalizable knots can be simply operated in geometry and explicitly calculated in algebra. In this paper we provide a universal form of renormalizable knots, which can be decomposed into a sequence of elementary templates; especially, it is independent of the traditional *-products of symbolic dynamics. We present some examples and list the elementary templates of period not bigger than 6.

Difference map and its electronic circuit realization

In this paper we study the dynamical behav- ior of the one-dimensional discrete-time system, the so-called iterated map. Namely, a bimodal quadratic map is introduced which is obtained as an amplifica- tion of the difference between well-known logistic and tent maps. Thus, it is denoted as the so-called differ- ence map. The difference map exhibits a variety of be- haviors according to the selection of the bifurcation parameter. The corresponding bifurcations are studied by numerical simulations and experimentally. The sta- bility of the difference map is studied by means of Lyapunov exponent and is proved to be chaotic ac- cording to Devaney's definition of chaos. Later on, a design of the electronic implementation of the differ- ence map is presented. The difference map electronic circuit is built using operational amplifiers, resistors and an analog multiplier. It turns out that this elec- tronic circuit presents fixed points, periodicity, chaos and intermittency that match with high accuracy to the corresponding values predicted theoretically.

SYNCHRONIZATION AND BASINS OF SYNCHRONIZED STATES IN TWO-DIMENSIONAL PIECEWISE MAPS VIA COUPLING THREE PIECES OF ONE-DIMENSIONAL MAPS

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

Invariants for the topological characterization of the iteration of differentiable functions – the bimodal case

We consider dynamical systems defined by a particular class of differentiable functions, as fixed state space. The dynamics is given by the iteration of an operator induced by a polynomial map which belongs to an appropriate family of isentropic bimodal interval maps. We characterize topologically these dynamical systems, in particular using the invariants defined for the iteration of the bimodal interval maps.