Complex Parameter Space Research Articles

Blending liquids

We present a method for smoothly blending between existing liquid animations. We introduce a semi-automatic method for matching two existing liquid animations, which we use to create new fluid motion that plausibly interpolates the input. Our contributions include a new space-time non-rigid iterative closest point algorithm that incorporates user guidance, a subsampling technique for efficient registration of meshes with millions of vertices, and a fast surface extraction algorithm that produces 3D triangle meshes from a 4D space-time surface. Our technique can be used to instantly create hundreds of new simulations, or to interactively explore complex parameter spaces. Our method is guaranteed to produce output that does not deviate from the input animations, and it generalizes to multiple dimensions. Because our method runs at interactive rates after the initial precomputation step, it has potential applications in games and training simulations.

A spectral synthesis code for rapid modelling of supernovae

We present TARDIS - an open-source code for rapid spectral modelling of supernovae (SNe). Our goal is to develop a tool that is sufficiently fast to allow exploration of the complex parameter spaces of models for SN ejecta. This can be used to analyse the growing number of high-quality SN spectra being obtained by transient surveys. The code uses Monte Carlo methods to obtain a self-consistent description of the plasma state and to compute a synthetic spectrum. It has a modular design to facilitate the implementation of a range of physical approximations that can be compared to asses both accuracy and computational expediency. This will allow users to choose a level of sophistication appropriate for their application. Here, we describe the operation of the code and make comparisons with alternative radiative transfer codes of differing levels of complexity (SYN++, PYTHON, and ARTIS). We then explore the consequence of adopting simple prescriptions for the calculation of atomic excitation, focussing on four species of relevance to Type Ia supernova spectra - Si II, S II, Mg II, and Ca II. We also investigate the influence of three methods for treating line interactions on our synthetic spectra and the need for accurate radiative rate estimates in our scheme.

Complex 2D matrix model and geometrical map on the complex-Nc plane

We study the parameter dependence of the internal structure of resonance states by formulating a complex two-dimensional (2D) matrix model, where the two dimensions represent two levels of resonances. We calculate a critical value of the parameter at which a “nature transition” with character exchange occurs between two resonance states, from the viewpoint of geometry on complex-parameter space. Such a critical value is useful for identifying the internal structure of resonance states with variation of the parameter in the system. We apply the model to analyze the internal structure of hadrons with variation of the color number |$N_c$| from |$\infty $| to a realistic value 3. By regarding |$1/N_c$| as the variable parameter in our model, we calculate a critical color number of the nature transition between hadronic states in terms of a quark–antiquark pair and a mesonic molecule as exotics from the geometry on the complex-|$N_c$| plane. For large-|$N_c$| effective theory, we employ the chiral Lagrangian induced by holographic QCD with a D4/D8/|$\overline {\mathrm {D8}}$| multi-D brane system in type IIA superstring theory.

When the World Changes in Your Hands: Similarity Ratings of Objects Morphing during Active Exploration

View-based theories of object recognition posit that coherent object representations are formed by linking together successive views of an actively explored object. This linking process relies on the assumption that the object does not change during exploration. Here, we test how object representations might be influenced when the shape of the object changes slowly during exploration. In our experiment, participants rated the similarity of two novel, 3D objects, whose shape was parametrically defined. Seventeen participants explored each object for 10 sec on an iPad which afforded natural and efficient interaction. The experiment contained a baseline condition, in which two objects of varying parameter-differences were presented, and a morphing condition, in which the first of the two objects slowly morphed during active exploration, making the objects more similar. Interestingly, no participant was aware of this morphing manipulation. Comparing baseline and morph trials, however, we found significantly higher similarity ratings during morphing [F(1,16) = 84.79, p < .001]. Furthermore, correlations between similarity ratings and differences in object parameters were high for the baseline condition (r = −.64), with smaller parameter differences being perceived as more similar. Interestingly, in the morphing condition correlations were lower for parameter differences after the morph (r = −.22), but remained high for differences before (r = −.47) and during morphing (r = −.50). In conclusion, similarity ratings in the baseline condition captured the complex parameter space well. Although participants did not notice the changing shape, morphing did systematically bias the ratings. Interestingly, similarity judgments correlated better in the initial exploration phase, suggesting a capacity limit for view integration of complex shapes.

Triangular spiral tilings

The topology of spiral tilings is intimately related to phyllotaxis theory and continued fractions. A quadrilateral spiral tiling is determined by a suitable chosen triple (ζ, m, n), where , and m and n are relatively prime integers. We give a simple characterization when (ζ, m, n) produce a triangular spiral tiling. When m and n are fixed, the admissible generators ζ form a curve in the unit disk. The family of triangular spiral tilings with opposed parastichy pairs (m, n) is parameterized by the divergence angle arg (ζ), while triangular spiral tilings with non-opposed parastichy pairs are parameterized by the plastochrone ratio 1/|ζ|. The generators for triangular spiral tilings with opposed parastichy pairs are not dense in the complex parameter space, while those with non-opposed parastichy pairs are dense. The proofs will be given in a general setting of spiral multiple tilings. We present paper-folding (origami) sheets that build spiral towers whose top-down views are triangular tilings.

Metabifurcation analysis unveils hidden dynamical structure of a neural population model

In a forthcoming publication [1], we find links between the physiological parameters and dynamics of a neural population model (NPM), which are not accessible by conventional methods. They emerge only when parameter space is systematically partitioned according to bifurcation responses, a method we call “metabifurcation analysis”. This is made possible by an automated bifurcation analysis of a large sample of admissible parameter sets and generalizes straightforwardly to the investigation of other models with large and complex parameter spaces. We study concretely the Liley NPM, which requires 29 different - biologically meaningful - parameters to be specified for bulk activity. In Ref. [2], 73,454 parameter sets were found that generate EEG power spectra predictions closely resembling those of human wakeful rest (“alpha sets”) at physiological neural firing rates. However, like many biologically realistic models (e.g., the Jansen-Rit NPM [3]), the Liley NPM supports a rich variety of dynamics which are generally distributed “fractally” over parameter space. Consequently, “alpha sets” are spread seemingly randomly throughout parameter space and hence a systematic classification of their dynamics and physiological correlations proves difficult. By studying bifurcation diagrams of a sub-sample of 405 “alpha sets”, we discovered that without exception all of them could be classified into two distinct dynamical families with regards to a continuation in R, governing overall inhibitory synaptic strength, and k, the relative effect of inhibition on excitatory vs. inhibitory subpopulations. Essentially, Family 1 (F1) has two almost parallel saddle-node lines in the R-k plane, whereas Family 2 (F2) sports two cusp points organizing two saddle-node wedges. We then sorted all 73,454 “alpha sets” into F1 and F2, respectively, by an automated procedure using AUTO-07P [4] that examined these distinctive features. Four important results were obtained by this bifurcation partitioning: 1) Reactions to the modeled application of the anesthetic agent isoflurane differ, with F2 being three times as likely as F1 to show strong, transient spectral power increases. 2) Exogenous stimuli modeled by thalamic input can control transitions between the dynamic families. 3) Distributions of the values of two excitatory population parameters differed strongly between F1 and F2, providing the potential for endogenous control. 4) The repertoire of (multi)stable oscillations for variations of inhibitory strength R was much more extensive for F2. In summary, we showcase here a new method for partitioning the complex parameter spaces of biologically realistic models according to their bifurcation behavior: “metabifurcation analysis”. For the Liley NPM, we find that one can classify “alpha sets” into two dynamical families that one can speculatively consider as representing “explorative” (F2, multistable oscillatory dynamics) and “consolidating” (F1, simple noise filter dynamics) brain states, where transitions between F1 and F2 are possible via exogenous stimuli.

Automated generation of compartmental models via database tools for neurophysiology data management, analysis, and simulation

Scientific progress depends increasingly on collaborative efforts that involve exchange of data and re-analysis of previously recorded data. A major obstacle to fully exploit the scientific potential of experimental data is the effort it takes to access both data and metadata for application of specific analysis methods, for exchange with collaborators, or for further analysis some time after the initial study was completed. To cope with these challenges and to make data analysis, re-analysis, and data sharing efficient, data together with metadata should be managed and accessed in a unified and reproducible way, so that the researcher can concentrate on the scientific questions rather than on problems of data management. At the German Neuroinformatics Node (http://www. g-node.org), an infrastructure for cellular and systems neuroscience is being developed to improve key ingredients of neuroscientific research: data access, data storage and exchange, and data analysis [1]. One central component is a data management platform where neuroscientists can store and organize their data for sharing and analysis. Tools and interfaces for data access through a variety of applications are being developed. They enable seamless integration of data access into the laboratory workflow and efficient management and selection of data in a systematic and largely automatized fashion for data sharing and analysis. To further demonstrate the applicability of this approach, we extended this system to generate compartmental models from the data stored in the database. Specifically, we investigated the effect of morphology on the electrophysiological properties of medial superior olive (MSO) neurons using morphology data from reconstructed neurons [2]. Simulating neurons from morphological data using compartmental models implies a highly complex parameter space, both in terms of the simulation parameters and the results. Our integrative approach employs a database to manage the data, control the simulations, and analyze the results. Data were stored and simulations were controlled via a PostgreSQL Database. To perform simulations, we used the NEURON simulator software [3], which was integrated via Python code using plpython. Automated simulations with different morphologies were triggered within the database. We used detailed models of reconstructed MSO neurons and, for comparison, simplified models with three compartments using the same passive parameters that were estimated with the complex models. Comparison of the simulated responses of the simple models to responses of the morphologically complex models showed that the shape of the neuron affected functional properties of the neurons, but that single geometric measures like surface area are not enough to explain specific results. Therefore, we suggest to use simulated electrophysiological measures like input resistance, or effective membrane time constant to map a complex morphology to a simplified n-compartmental model. Our study demonstrates the efficiency of a unifying approach employing a database for management of data, analysis, and simulation tools. This in turn facilitates analysis and visualization of results, which can be presented in a portable and accessible way via direct database access and web interface.

Metabifurcation analysis of a mean field model of the cortex

Mean field models (MFMs) of cortical tissue incorporate salient, average features of neural masses in order to model activity at the population level, thereby linking microscopic physiology to macroscopic observations, e.g., with the electroencephalogram (EEG). One of the common aspects of MFM descriptions is the presence of a high-dimensional parameter space capturing neurobiological attributes deemed relevant to the brain dynamics of interest. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two archetypal categories or “families”. After investigating and characterizing them in depth, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli such as thalamic input, and distributions of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs significantly between families, we are able to show how metamorphoses between the families can be brought about by exogenous stimuli. We here unveil links between measurable physiological attributes of the brain and dynamical patterns that are not accessible by linear methods. They instead emerge when the nonlinear structure of parameter space is partitioned according to bifurcation responses. We call this general method “metabifurcation analysis”. The partitioning cannot be achieved by the investigation of only a small number of parameter sets and is instead the result of an automated bifurcation analysis of a representative sample of 73,454 physiologically admissible parameter sets. Our approach generalizes straightforwardly and is well suited to probing the dynamics of other models with large and complex parameter spaces.

Visual and haptic perceptual spaces show high similarity in humans

In this study, we show that humans form highly similar perceptual spaces when they explore complex objects from a parametrically defined object space in the visual and haptic domains. For this, a three-dimensional parameter space of well-defined, shell-like objects was generated. Participants either explored two-dimensional pictures or three-dimensional, interactive virtual models of these objects visually, or they explored three-dimensional plastic models haptically. In all cases, the task was to rate the similarity between two objects. Using these similarity ratings and multidimensional scaling (MDS) analyses, the perceptual spaces of the different modalities were then analyzed. Looking at planar configurations within this three-dimensional object space, we found that active visual exploration led to a highly similar perceptual space compared to passive exploration, showing that participants were able to reconstruct the complex parameter space already from two-dimensional pictures alone. Furthermore, we found that visual and haptic perceptual spaces had virtually identical topology compared to that of the physical stimulus space. Surprisingly, the haptic modality even slightly exceeded the visual modality in recovering the topology of the complex object space when the whole three-dimensional space was explored. Our findings point to a close connection between visual and haptic object representations and demonstrate the great degree of fidelity with which haptic shape processing occurs.

Breakdown of invariant attractors for the dissipative standard map

We implement different methods for the computation of the breakdown threshold of invariant attractors in the dissipative standard mapping. A first approach is based on the computation of the Sobolev norms of the function parametrizing the solution. Then we look for the approximating periodic orbits and we analyze their stability in order to compute the critical threshold at which an invariant attractor breaks down. We also determine the domain of convergence of the dissipative standard mapping by extending the computations to the complex parameter space as well as by investigating a two-frequency model.

The potential of microfluidic water-in-oil droplets in experimental biology

The comprehensive characterisation of complex parameter space in '-omics' technologies requires high-throughput systems. In vitro compartmentalisation of reactions in water-in-oil droplets combines the necessary ability to carry out large numbers of experiments under controlled conditions with quantitative readout, and has recently advanced towards automation by generating droplets in microfluidic devices. Some approaches based on these principles are already familiar (e.g. emulsion PCR for sequencing), others, including directed evolution or cell-based assays, are in advanced stages of development--and proof-of-principle experiments are appearing for a whole range of applications in diagnostics, cellomics, proteomics, drug discovery and systems and synthetic biology. This review describes the current state-of-the-art, notes salient features of successful experiments and extrapolates in the direction of more highly integrated systems.

A transdimensional Bayesian model for pattern recognition in DNA sequences

Identification of transcription factor binding sites (TFBSs) is essential to elucidate gene regulatory networks. This article is focused on the recognition of overpresented short patterns, called "motifs", that may correspond to regulatory binding sites in the DNA sequences upstream of genes. An integrated Bayesian model is proposed to incorporate all unknown characteristics in motif discovery, including the number of motifs, motif widths, motif compositions, the number of motif sites, and locations of motif sites. Reversible jump Markov chain Monte Carlo is used to obtain posterior inference in the transdimensional parameter space. We present a number of suggestions for graphical summarization of the posterior distribution over the complex parameter space. The basic model is extended using a third-order Markov structure for nonmotif bases and allowing positions within a motif to be switched between 2 types: "conserved" and "degenerate." We evaluate the prediction accuracy for the simulated data with 3 motifs and apply the model to upstream sequences in high signal-to-noise regions in a human ChIP-chip study. The performance of the Bayesian model is assessed using yeast data sets of various numbers of sequences and background structures, with and without true TFBSs. The performance is also compared to other computational methods, including 2 statistical approaches, AlignACE and multiple expectation maximization for motif elicitation, and 1 word numeration-based approach, yeast motif finder (YMF).

Measuring supersymmetry

If new physics is found at the LHC (and the ILC) the reconstruction of the underlying theory should not be biased by assumptions about high-scale models. For the mapping of many measurements onto high-dimensional parameter spaces we introduce SFitter with its new weighted Markov chain technique. SFitter constructs an exclusive likelihood map, determines the best-fitting parameter point and produces a ranked list of the most likely parameter points. Using the example of the TeV-scale supersymmetric Lagrangian we show how a high-dimensional likelihood map will generally include degeneracies and strong correlations. SFitter allows us to study such model-parameter spaces employing Bayesian as well as frequentist constructions. We illustrate in detail how it should be possible to analyze high-dimensional new-physics parameter spaces like the TeV-scale MSSM at the LHC. A combination of LHC and ILC measurements might well be able to completely cover highly complex TeV-scale parameter spaces.

The HARPS search for southern extra-solar planets

Context. The µ Arae planetary system is fairly complex, because it contains two already known planets, µ Arae b with P = 640 days and µ Arae c with P = 9.64 days , and a third companion on a wide, but still poorly defined, orbit. Aims. Even with three planets in the system, the data points keep anomalously high dispersion around the fitted solution. The high residuals are only partially due to the strong p-mode oscillations of the host star. We therefore studied the possible presence of a fourth planet in this system. Methods. During the past years we carried out additional and extremely precise radial-velocity measurements with the HARPS spectrograph. These data turned out to be highly important for constraining the many free parameters in a four-planet orbital fit. Nevertheless, the search for the best solution remains difficult in this complex and multi-dimensional parameter space. The new Stakanof software, that uses an optimized genetic algorithm, helped us considerably in this task and made our search extremely efficient and successful. Results. We provide a full orbital solution of the planetary system around µ Arae. It turns out to be the second system known to harbor 4 planetary companions. Before this study, µ Arae b was already well known and characterized. Thanks to the new data points acquired with HARPS we can confirm the presence of µ Arae c at P = 9.64 days, which produces a coherent RV signal over more than two years. The new orbital fit sets the mass of µ Arae c to 10.5 M⊕. Furthermore, we present the discovery of µ Arae d, a new planet on an almost circular 310 day-period and with a mass of 0.52 MJup. Finally, we give completely new orbital parameters for the longest-period planet, µ Arae e. It is the first time that this companion has been constrained by radial-velocity data into a dynamical stable orbit, which leaves no doubt about its planetary nature. We take this opportunity to discuss naming conventions for poorly characterized planets.

Quantum criticality at the infinite complete basis set limit: A thermodynamic analog of the Yang and Lee theorem

Finite size scaling for calculations of the critical parameters of the few-body Schrödinger equation is based on taking the number of elements in a complete basis set as the size of the system. We show in an analogy with Yang and Lee theorem, which states that singularities of the free energy at phase transitions occur only in the thermodynamic limit, that singularities in the ground state energy occur only in the infinite complete basis set limit. To illustrate this analogy in the complex-parameter space, we present calculations for Yukawa type potential, and a Coulomb type potential for two-electron atoms.

Lee-Yang zero analysis for the study of QCD phase structure

We comment on the Lee-Yang zero analysis for the study of the phase structure of QCD at high temperature and baryon number density by Monte-Carlo simulations. We find that the sign problem for non-zero density QCD induces a serious problem in the finite volume scaling analysis of the Lee-Yang zeros for the investigation of the order of the phase transition. If the sign problem occurs at large volume, the Lee-Yang zeros will always approach the real axis of the complex parameter plane in the thermodynamic limit. This implies that a scaling behavior which would suggest a crossover transition will not be obtained. To clarify this problem, we discuss the Lee-Yang zero analysis for SU(3) pure gauge theory as a simple example without the sign problem, and then consider the case of non-zero density QCD. It is suggested that the distribution of the Lee-Yang zeros in the complex parameter space obtained by each simulation could be more important information for the investigation of the critical endpoint in the $(T, \mu_q)$ plane than the finite volume scaling behavior.

Surface diffraction structure determination from combinatorial simultaneous optimization

We present a technique to efficiently locate the global minimum of a cost function on a complex multi-dimensional parameter space (e.g. a structural reliability R-factor). The method builds a convergent series of structures from combinatorial simultaneous optimization of all the parameters on different subspaces of the experimental data base. Performance has been tested for two model situations: (1) phase retrieval from electron scattering by a single atom and (2) low-energy electron diffraction analysis of simulated I( V) curves for the Ir(1 1 0)-p(2 × 1) missing row. Compared under the same conditions, a reduction in the computation effort, is found w.r.t. previous state-of-the-art methods (e.g., simulated annealing and genetic algorithms).

Geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces

We investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form 〈ψf|ψ〉, where |ψf〉 is a fixed state in Hilbert space and |ψ〉 is a variable state, treated as a section of a U(1) bundle over a complex subspace of the corresponding ray space R=CPn. Amplitude functions on such holomorphic line bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy–Riemann conditions involving the U(1) Berry–Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulus, therefore allowing for the reconstruction of the phase from the modulus (or vice versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space R=CPn, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini–Study metric. In conjunction with the generalized Cauchy–Riemann conditions, this geodesic interpretation leads to additional relations, in particular, a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.

Optimization of Multiple Freshwater Diversions in Well-Mixed Estuaries

A method is presented to optimize diversions of freshwater from multiple points in a well-mixed estuary. The objective is to prevent salinity concentrations at a point in the brackish water zone from exceeding levels that are stipulated for environmental defense. Optimization is achieved using an efficient gradient-based search algorithm (quasi-Newton with Broyden-Fletcher-Goldfarb-Shanno update). Time-invariant and time-variant diversion rates are considered as control parameters, and time-variant diversion rates are shown to permit larger diversions overall. The latter requires the identification of a vastly larger number of parameters in a more complex parameter space. Computations are carried out using a 1D model formulation applied to a series of hypothetical systems and the St. Jones estuary in Delaware.

The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms

The vacuum structure and spectra of two-dimensional gauge theories with N=(2,2) supersymmetry are investigated. These theories admit a twisted mass term for charged chiral matter multiplets. In the case of a U(1) gauge theory with N chiral multiplets of equal charge, an exact description of the BPS spectrum is obtained for all values of the twisted masses. The BPS spectrum has two dual descriptions which apply in the Higgs and Coulomb phases of the theory respectively. The two descriptions are related by a massive analog of mirror symmetry: the exact mass formula which is given by a one-loop calculation in the Coulomb phase gives predictions for an infinite series of instanton corrections in the Higgs phase. The theory is shown to exhibit many phenomena which are usually associated with N=2 theories in four dimensions. These include BPS-saturated dyons which carry both topological and Noether charges, non-trivial monodromies of the spectrum in the complex parameter space, curves of marginal stability on which BPS states can decay and strongly coupled vacua with massless solitons and dyons.