Period-doubling/symmetry-breaking mode interactions in iterated maps

We consider iterated maps with a reflectional symmetry. Possible bifurcations in such systems include period-doubling bifurcations (within the symmetric subspace) and symmetry-breaking bifurcations. By using a second parameter, these bifurcations can be made to coincide at a mode interaction. By reformulating the period-doubling bifurcation as a symmetry-breaking bifurcation, two bifurcation equations with Z₂×Z₂ symmetry are derived. A local analysis of solutions is then considered, including the derivation of conditions for a tertiary Hopf bifurcation. Applications to symmetrically coupled maps and to two coupled, vertically forced pendulums are described.     

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially, this work can be used for some real applications in secure communication, such as data and image encryptions.     

Dynamic maps in phase-conjugated optical resonators

In this article the dynamical behavior of a beam within a ring-phase-conjugated resonator is modeled using two-dimensional iterative maps. In particular and as an example it is explicitly shown how the difference equations of the Duffing map can be used to describe the dynamic behavior of what we call Duffing beams i.e. beams that behave according to the Duffing map. The matrix of a Duffing map generating device is found in terms of the Duffing parameters, the state variables and the resonator parameters. To our knowledge this is the first time that the mathematical characteristics of an optical device in an optical cavity are stated so that a Duffing map is obtained as the dynamics for the ray beams.     

Stochastic resonance and chaotic resonance in bimodal maps: A case study

We present the results of an extensive numerical study on the phenomenon of stochastic resonance in a bimodal cubic map. Both Gaussian random noise as well as deterministic chaos are used as input to drive the system between the basins. Our main result is that when two identical systems capable of stochastic resonance are coupled, the SNR of either system is enhanced at an optimum coupling strength. Our results may be relevant for the study of stochastic resonance in biological systems.     

Period matching in modulated maps

We study discrete nonlinear maps in which the control parameter is itself “modulated” by another discrete nonlinear map. We show that for a certain class of such maps, which includes for example the logistic map, the periodicity of the modulated signal is either one, independent of the periodicity of the modulating signal, or its periodicity is an integral multiple of the periodicity of the modulating signal or it is chaotic.     

Improved temporal clustering analysis method applied to whole-brain data in acupuncture fMRI study

Temporal clustering analysis (TCA) has been proposed as a method for detecting the brain responses of a functional magnetic resonance imaging (fMRI) time series when the time and location of activation are completely unknown. But TCA is not suitable for treating the time series of the whole brain due to the existence of many inactive pixels. In theory, active pixels are located only in gray matter (GM). In this study, SPM2 was used to segment functional images into GM, white matter and cerebrospinal fluid, and only the pixels in GM were considered. Thus, most of inactive pixels are deleted, so that the sensitivity of TCA is greatly improved in the analysis of the whole brain. The same set of acupuncture fMRI data was treated using both conventional TCA and modified TCA (MTCA) for comparing their analytical ability. The results clearly show a significant improvement in the sensitivity achieved by MTCA.     

Initial attempts at directly detecting alpha wave activity in the brain using MRI

The “direct detection” of neuronal activity by MRI could offer improved spatial and temporal resolution compared to the blood oxygenation level-dependent (BOLD) effect. Here we describe initial attempts to use MRI to detect directly the neuronal currents resulting from spontaneous alpha wave activity, which have previously been shown to generate the largest extracranial magnetic fields. Experiments were successfully carried out on four subjects at 3 T. A single slice was imaged at a rate of 25 images per second under two conditions. The first (in darkness with eyes-closed) was chosen to promote alpha wave activity, while the second (eyes-open viewing a visual stimulus) was chosen to suppress it. The fluctuations of the phase and magnitude of the resulting MR image data were frequency analysed, and tested for the signature of both alpha wave activity and neuronal activity evoked by the visual stimulus.

Regions were found that consistently showed elevated power in fluctuations of the phase of the MR signal, in the frequency range of alpha waves, during the eyes-closed condition. It was conservatively assumed that if oscillations occurred at the same frequency in the magnitude signal from the same region or at the same frequency in the phase or magnitude signal from other regions overlying large vessels or cerebrospinal fluid (CSF), then the phase changes were not due to neuronal activity related to alpha waves. Using these criteria the data obtained were consistent with direct detection of alpha wave activity in three of the four volunteers. No significant MR signal fluctuations due to evoked activity were identified.     

Community structure and modularity in networks of correlated brain activity

Functional connectivity patterns derived from neuroimaging data may be represented as graphs or networks, with individual image voxels or anatomically-defined structures representing the nodes, and a measure of correlation between the responses in each pair of nodes determining the edges. This explicit network representation allows network-analysis approaches to be applied to the characterization of functional connections within the brain. Much recent research in complex networks has focused on methods to identify community structure, i.e. cohesive clusters of strongly interconnected nodes. One class of such algorithms determines a partition of a network into 'sub-networks' based on the optimization of a modularity parameter, thus also providing a measure of the degree of segregation versus integration in the full network. Here, we demonstrate that a community structure algorithm based on the maximization of modularity, applied to a functional connectivity network calculated from the responses to acute fluoxetine challenge in the rat, can identify communities whose distributions correspond to anatomically meaningful structures and include compelling functional subdivisions in the brain. We also discuss the biological interpretation of the modularity parameter in terms of segregation and integration of brain function.     

Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps

Bifurcation structures for nonlinear dynamical systems in a space of two parameters often display geometric shapes resembling shrimps. For one-dimensional maps with two parameters and multiple extrema, the underlying structure of the shrimps can be elucidated by computing the locus of superstable cycles which form a “skeleton” that supports the shrimps. Here we use continuation methods to identify and compute structures in two-dimensional maps that play the same role as the skeleton in one-dimensional maps. This facilitates determining the complex geometries for situations in which there is multistability, and for which the regions of parameter space supporting stable orbits get vanishingly small.     

The structure of mode-locked regions in quasi-periodically forced circle maps

Using a mixture of analytic and numerical techniques we show that the mode-locked regions of quasi-periodically forced Arnold circle maps form complicated sets in parameter space. These sets are characterized by ‘pinched-off’ regions, where the width of the mode-locked region becomes very small. By considering general quasi-periodically forced circle maps we show that this pinching occurs in a broad class of such maps having a simple symmetry.     

A study on small-world brain functional networks altered by postherpetic neuralgia

Understanding the effect of postherpetic neuralgia (PHN) pain on brain activity is important for clinical strategies. This is the first study, to our knowledge, to relate PHN pain to small-world properties of brain functional networks. Functional magnetic resonance imaging (fMRI) was used to construct functional brain networks of the subjects during the resting state. Sixteen patients with PHN pain and 16 (8 males, 8 females for both groups) age-matched controls were studied. The PHN patients exhibited decreased local efficiency along with non-significant changes of global efficiency in comparison with the healthy controls. Moreover, regional nodal efficiency was found to be significantly affected by PHN pain in the areas related to sense (postcentral gyrus, inferior parietal gyrus and thalamus), memory/affective processes (parahippocampal gyrus) and emotional activities (putamen). Significant correlation (p < 0.05) was also found between the nodal efficiency of putamen and pain intensity in PHN patients. Our results suggest that PHN modulates the local efficiency, and the small-world properties of brain networks may have potentials to objectively evaluate pain information in clinic.     

Finite data-size scaling of clustering in earthquake networks

An earthquake network is known to be of the small-world type. The values of the network’s characteristics, however, depend not only on the cell size (i.e., the scale of coarse graining needed for constructing the network) but also on the size of a seismic data set. Here, discovery of a scaling law for the clustering coefficient in terms of the data size, which is referred to here as finite data-size scaling, is reported. Its universality is shown to be supported by the detailed analysis of the data taken from California, Japan and Iran. Effects of setting a threshold of magnitude are also discussed.     

Helium clustering in neutron-rich Be isotopes

Measurements of the helium-cluster breakup and neutron removal cross-sections for neutron-rich Be isotopes ^10–12,14Be are presented. These have been studied in the 30 to 42 MeV/nucleon energy range where reaction measurements are proposed to be sensitive to the cluster content of the ground-state wave-function. These measurements provide a comprehensive survey of the decay processes of the Be isotopes by which the valence neutrons are removed revealing the underlying – core-cluster structure. The measurements indicate that clustering in the Be isotopes remains important up to the drip-line nucleus ¹⁴Be and that the dominant helium-cluster structure in the neutron-rich Be isotopes corresponds to –Xn–.     

An improved temporal clustering analysis method applied to whole-brain data in fMRI study

Temporal clustering analysis (TCA) has been proposed as a method to detect the brain responses of an fMRI time series when the time and location of the activation are completely unknown. But TCA is still incompetent in dealing with the time series of the whole brain due to the existence of many inactive pixels. If only active pixels are considered, the sensitivity of TCA will be improved greatly and it could be applied to the whole brain. In this study, some modifications were made to TCA to remove inactive pixels, and the applicability of the modified TCA to the whole brain was validated with a set of visual fMRI data. Based on the time series of the modified TCA, activations of the whole brain corresponding to the visual stimulation were detected. Compared with the previous TCA, the modified TCA method shows a significant improvement in the sensitivity to detect activation peaks of the whole brain.     

Effective suppression of period-doubling in two diffusively coupled logistic maps

We consider a system of two delay diffusively coupled logistic maps. We find that for moderate values of diffusion coupling, the period-doubling sequence is effectively suppressed. Our study supports the existence of certain generic features for systems consisting of two coupled maps.     

Characterizing turbulence in globally coupled maps with stochastic finite automata

We describe a method to discriminate between ordered and turbulent behavior in a general class of collective systems known as Globally Coupled Maps (GCM). Our method is able to discover an unknown small ordered region inside the turbulent phase of GCM parameter space. The computational nature of the method is the main novelty of our approach; it is another example of how measures based on computational notions of structure may provide new information in the study of dynamical systems.     

Long-duration transcutaneous electric acupoint stimulation alters small-world brain functional networks

Acupuncture, which is recognized as an alternative and complementary treatment in Western medicine, has long shown efficiencies in chronic pain relief, drug addiction treatment, stroke rehabilitation and other clinical practices. The neural mechanism underlying acupuncture, however, is still unclear. Many studies have focused on the sustained effects of acupuncture on healthy subjects, yet there are very few on the topological organization of functional networks in the whole brain in response to long-duration acupuncture (longer than 20 min). This paper presents a novel study on the effects of long-duration transcutaneous electric acupoint stimulation (TEAS) on the small-world properties of brain functional networks. Functional magnetic resonance imaging was used to construct brain functional networks of 18 healthy subjects (9 males and 9 females) during the resting state. All subjects received both TEAS and minimal TEAS (MTEAS) and were scanned before and after each stimulation. An altered functional network was found with lower local efficiency and no significant change in global efficiency for healthy subjects after TEAS, while no significant difference was observed after MTEAS. The experiments also showed that the nodal efficiencies in several paralimbic/limbic regions were altered by TEAS, and those in middle frontal gyrus and other regions by MTEAS. To remove the psychological effects and the baseline, we compared the difference between diffTEAS (difference between after and before TEAS) and diffMTEAS (difference between after and before MTEAS). The results showed that the local efficiency was decreased and that the nodal efficiencies in frontal gyrus, orbitofrontal cortex, anterior cingulate gyrus and hippocampus gyrus were changed. Based on those observations, we conclude that long-duration TEAS may modulate the short-range connections of brain functional networks and also the limbic system.     

Non-integrability vs. integrability in pentagram maps

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.     

Detecting functional connectivity in the resting brain: a comparison between ICA and CCA

Independent component analysis (ICA) and cross-correlation analysis (CCA) are general tools for detecting resting-state functional connectivity. In this study, we jointly evaluated these two approaches based on simulated data and in vivo functional magnetic resonance imaging data acquired from 10 resting healthy subjects. The influence of the number of independent components (maps) on the results of ICA was investigated. The influence of the selection of the seeds on the results of CCA was also examined. Our results reveal that significant differences between these two approaches exist. The performance of ICA is superior as compared with that of CCA; in addition, the performance of ICA is not significantly affected by structured noise over a relatively large range. The results of ICA could be affected by the number of independent components if this number is too small, however. Converting the spatially independent maps of ICA into z maps for thresholding tends to overestimate the false-positive rate. However, the overestimation is not very severe and may be acceptable in most cases. The results of CCA are dependent on seeds location. Seeds selected based on different criteria will significantly affect connectivity maps.