Nonlinear dynamic transfer functions for certain retinal neuronal systems

The applicability of the white-noise method to the identification of a nonlinear system is investigated. Subsequently, the method is applied to certain vertebrate retinal neuronal systems and nonlinear, dynamic transfer functions are derived which describe quantitatively the information transformations starting with the light-pattern stimulus and culminating in the ganglion response which constitutes the visually-derived input to the brain. The retina of the catfish, Ictalurus punctatus, is used for the experiments.

The Wiener formulation of the white-noise theory is shown to be impractical and difficult to apply to a physical system. A different formulation based on crosscorrelation techniques is shown to be applicable to a wide range of physical systems provided certain considerations are taken into account. These considerations include the time-invariancy of the system, an optimum choice of the white-noise input bandwidth, nonlinearities that allow a representation in terms of a small number of characterizing kernels, the memory of the system and the temporal length of the characterizing experiment. Error analysis of the kernel estimates is made taking into account various sources of error such as noise at the input and output, bandwidth of white-noise input and the truncation of the gaussian by the apparatus.

Nonlinear transfer functions are obtained, as sets of kernels, for several neuronal systems: Light → Receptors, Light → Horizontal, Horizontal → Ganglion, Light → Ganglion and Light → ERG. The derived models can predict, with reasonable accuracy, the system response to any input. Comparison of model and physical system performance showed close agreement for a great number of tests, the most stringent of which is comparison of their responses to a white-noise input. Other tests include step and sine responses and power spectra.

Many functional traits are revealed by these models. Some are: (a) the receptor and horizontal cell systems are nearly linear (small signal) with certain "small" nonlinearities, and become faster (latency-wise and frequency-response-wise) at higher intensity levels, (b) all ganglion systems are nonlinear (half-wave rectification), (c) the receptive field center to ganglion system is slower (latency-wise and frequency-response-wise) than the periphery to ganglion system, (d) the lateral (eccentric) ganglion systems are just as fast (latency and frequency response) as the concentric ones, (e) (bipolar response) = (input from receptors) - (input from horizontal cell), (f) receptive field center and periphery exert an antagonistic influence on the ganglion response, (g) implications about the origin of ERG, and many others.

An analytical solution is obtained for the spatial distribution of potential in the S-space, which fits very well experimental data. Different synaptic mechanisms of excitation for the external and internal horizontal cells are implied.

Random propagation in complex systems : nonlinear matrix recursions and epidemic spread

This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.

Nonlinear spectroscopy of cold atoms in diffuse laser light

The nonlinear spectroscopy of cold atoms in the diffuse laser cooling system is studied in this paper. We present the theoretical models of the recoil-induced resonances (RIR) and the electromagnetically-induced absorption (EIA) of cold atoms in diffuse laser light, and show their signals in an experiment of cooling Rb-87 atomic vapor in an integrating sphere. The theoretical results are in good agreement with the experimental ones when the light intensity distribution in the integrating sphere is considered. The differences between nonlinear spectra of cold atoms in the diffuse laser light and in the optical molasses are also discussed. (c) 2009 Optical Society of America

SteelConverter and Caltech VirtualShaker: Rapid Nonlinear Cloud-Based Structural Model Conversion and Analysis

STEEL, the Caltech created nonlinear large displacement analysis software, is currently used by a large number of researchers at Caltech. However, due to its complexity, lack of visualization tools (such as pre- and post-processing capabilities) rapid creation and analysis of models using this software was difficult. SteelConverter was created as a means to facilitate model creation through the use of the industry standard finite element solver ETABS. This software allows users to create models in ETABS and intelligently convert model information such as geometry, loading, releases, fixity, etc., into a format that STEEL understands. Models that would take several days to create and verify now take several hours or less. The productivity of the researcher as well as the level of confidence in the model being analyzed is greatly increased.

It has always been a major goal of Caltech to spread the knowledge created here to other universities. However, due to the complexity of STEEL it was difficult for researchers or engineers from other universities to conduct analyses. While SteelConverter did help researchers at Caltech improve their research, sending SteelConverter and its documentation to other universities was less than ideal. Issues of version control, individual computer requirements, and the difficulty of releasing updates made a more centralized solution preferred. This is where the idea for Caltech VirtualShaker was born. Through the creation of a centralized website where users could log in, submit, analyze, and process models in the cloud, all of the major concerns associated with the utilization of SteelConverter were eliminated. Caltech VirtualShaker allows users to create profiles where defaults associated with their most commonly run models are saved, and allows them to submit multiple jobs to an online virtual server to be analyzed and post-processed. The creation of this website not only allowed for more rapid distribution of this tool, but also created a means for engineers and researchers with no access to powerful computer clusters to run computationally intensive analyses without the excessive cost of building and maintaining a computer cluster.

In order to increase confidence in the use of STEEL as an analysis system, as well as verify the conversion tools, a series of comparisons were done between STEEL and ETABS. Six models of increasing complexity, ranging from a cantilever column to a twenty-story moment frame, were analyzed to determine the ability of STEEL to accurately calculate basic model properties such as elastic stiffness and damping through a free vibration analysis as well as more complex structural properties such as overall structural capacity through a pushover analysis. These analyses showed a very strong agreement between the two softwares on every aspect of each analysis. However, these analyses also showed the ability of the STEEL analysis algorithm to converge at significantly larger drifts than ETABS when using the more computationally expensive and structurally realistic fiber hinges. Following the ETABS analysis, it was decided to repeat the comparisons in a software more capable of conducting highly nonlinear analysis, called Perform. These analyses again showed a very strong agreement between the two softwares in every aspect of each analysis through instability. However, due to some limitations in Perform, free vibration analyses for the three story one bay chevron brace frame, two bay chevron brace frame, and twenty story moment frame could not be conducted. With the current trend towards ultimate capacity analysis, the ability to use fiber based models allows engineers to gain a better understanding of a building’s behavior under these extreme load scenarios.

Following this, a final study was done on Hall’s U20 structure [1] where the structure was analyzed in all three softwares and their results compared. The pushover curves from each software were compared and the differences caused by variations in software implementation explained. From this, conclusions can be drawn on the effectiveness of each analysis tool when attempting to analyze structures through the point of geometric instability. The analyses show that while ETABS was capable of accurately determining the elastic stiffness of the model, following the onset of inelastic behavior the analysis tool failed to converge. However, for the small number of time steps the ETABS analysis was converging, its results exactly matched those of STEEL, leading to the conclusion that ETABS is not an appropriate analysis package for analyzing a structure through the point of collapse when using fiber elements throughout the model. The analyses also showed that while Perform was capable of calculating the response of the structure accurately, restrictions in the material model resulted in a pushover curve that did not match that of STEEL exactly, particularly post collapse. However, such problems could be alleviated by choosing a more simplistic material model.

Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets

Jet noise reduction is an important goal within both commercial and military aviation. Although large-scale numerical simulations are now able to simultaneously compute turbulent jets and their radiated sound, lost-cost, physically-motivated models are needed to guide noise-reduction efforts. A particularly promising modeling approach centers around certain large-scale coherent structures, called wavepackets, that are observed in jets and their radiated sound. The typical approach to modeling wavepackets is to approximate them as linear modal solutions of the Euler or Navier-Stokes equations linearized about the long-time mean of the turbulent flow field. The near-field wavepackets obtained from these models show compelling agreement with those educed from experimental and simulation data for both subsonic and supersonic jets, but the acoustic radiation is severely under-predicted in the subsonic case. This thesis contributes to two aspects of these models. First, two new solution methods are developed that can be used to efficiently compute wavepackets and their acoustic radiation, reducing the computational cost of the model by more than an order of magnitude. The new techniques are spatial integration methods and constitute a well-posed, convergent alternative to the frequently used parabolized stability equations. Using concepts related to well-posed boundary conditions, the methods are formulated for general hyperbolic equations and thus have potential applications in many fields of physics and engineering. Second, the nonlinear and stochastic forcing of wavepackets is investigated with the goal of identifying and characterizing the missing dynamics responsible for the under-prediction of acoustic radiation by linear wavepacket models for subsonic jets. Specifically, we use ensembles of large-eddy-simulation flow and force data along with two data decomposition techniques to educe the actual nonlinear forcing experienced by wavepackets in a Mach 0.9 turbulent jet. Modes with high energy are extracted using proper orthogonal decomposition, while high gain modes are identified using a novel technique called empirical resolvent-mode decomposition. In contrast to the flow and acoustic fields, the forcing field is characterized by a lack of energetic coherent structures. Furthermore, the structures that do exist are largely uncorrelated with the acoustic field. Instead, the forces that most efficiently excite an acoustic response appear to take the form of random turbulent fluctuations, implying that direct feedback from nonlinear interactions amongst wavepackets is not an essential noise source mechanism. This suggests that the essential ingredients of sound generation in high Reynolds number jets are contained within the linearized Navier-Stokes operator rather than in the nonlinear forcing terms, a conclusion that has important implications for jet noise modeling.

Experimental study on a nonlinear photonics process of Er(0.5)Yb(3): FOV oxyfluoride nanophase vitroceramics

We study the nonlinear photonics of rare-earth-doped oxyfluoride nanophase vitroceramics (FOV), oxyfluoride glass (FOG), and ZBLAN fluoride glass. We found that an interesting fluorescence intensity inversion phenomenon between red and green fluorescence occurs from Er(0.5)Yb(3):FOV The dynamic range Sigma of the intensity inversion between red and green fluorescence of Er(0.5)Yb(3):FOV is about 5.753 x 10(2), which is 100 to 1000 times larger than those of other materials. One of the applications of this phenomenon is double-wavelength fluorescence falsification-preventing technology, which is proved to possess the novel antifriction loss and antiscribble properties. (c) 2007 Optical Society of America.

Learning the Structure of Deep Sparse Graphical Models

Deep belief networks are a powerful way to model complex probability distributions. However, learning the structure of a belief network, particularly one with hidden units, is difficult. The Indian buffet process has been used as a nonparametric Bayesian prior on the directed structure of a belief network with a single infinitely wide hidden layer. In this paper, we introduce the cascading Indian buffet process (CIBP), which provides a nonparametric prior on the structure of a layered, directed belief network that is unbounded in both depth and width, yet allows tractable inference. We use the CIBP prior with the nonlinear Gaussian belief network so each unit can additionally vary its behavior between discrete and continuous representations. We provide Markov chain Monte Carlo algorithms for inference in these belief networks and explore the structures learned on several image data sets.

Model based learning of sigma points in unscented Kalman filtering

The unscented Kalman filter (UKF) is a widely used method in control and time series applications. The UKF suffers from arbitrary parameters necessary for a step known as sigma point placement, causing it to perform poorly in nonlinear problems. We show how to treat sigma point placement in a UKF as a learning problem in a model based view. We demonstrate that learning to place the sigma points correctly from data can make sigma point collapse much less likely. Learning can result in a significant increase in predictive performance over default settings of the parameters in the UKF and other filters designed to avoid the problems of the UKF, such as the GP-ADF. At the same time, we maintain a lower computational complexity than the other methods. We call our method UKF-L. ©2010 IEEE.

An overview of Sequential Monte Carlo methods for parameter estimation in general state-space models

Nonlinear non-Gaussian state-space models arise in numerous applications in control and signal processing. Sequential Monte Carlo (SMC) methods, also known as Particle Filters, provide very good numerical approximations to the associated optimal state estimation problems. However, in many scenarios, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard SMC methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive overview of SMC methods that have been proposed to perform static parameter estimation in general state-space models. We discuss the advantages and limitations of these methods. © 2009 IFAC.

How are internal models of unstable tasks formed?

The results of recent studies suggest that humans can form internal models that they use in a feedforward manner to compensate for both stable and unstable dynamics. To examine how internal models are formed, we performed adaptation experiments in novel dynamics, and measured the endpoint force, trajectory and EMG during learning. Analysis of reflex feedback and change of feedforward commands between consecutive trials suggested a unified model of motor learning, which can coherently unify the learning processes observed in stable and unstable dynamics and reproduce available data on motor learning. To our knowledge, this algorithm, based on the concurrent minimization of (reflex) feedback and muscle activation, is also the first nonlinear adaptive controller able to stabilize unstable dynamics.

Characterization of dynamic nonlinear absorption of carbon nanotube saturable absorber

Dynamic nonlinear absorption of composite-type single-wall carbon nanotube saturable absorbers is characterized using both femtosecond and picosecond pump pulses. Results are compared with numerical simulations based on two commonly used saturable absorber models. © 2010 Optical Society of America.

Model based learning of sigma points in unscented Kalman filtering

The unscented Kalman filter (UKF) is a widely used method in control and time series applications. The UKF suffers from arbitrary parameters necessary for sigma point placement, potentially causing it to perform poorly in nonlinear problems. We show how to treat sigma point placement in a UKF as a learning problem in a model based view. We demonstrate that learning to place the sigma points correctly from data can make sigma point collapse much less likely. Learning can result in a significant increase in predictive performance over default settings of the parameters in the UKF and other filters designed to avoid the problems of the UKF, such as the GP-ADF. At the same time, we maintain a lower computational complexity than the other methods. We call our method UKF-L. © 2011 Elsevier B.V.

Linear and nonlinear model large-eddy simulations of a plane jet

The nonlinear Kosovic, and mixed Leray and α subgrid scale models are contrasted with linear Smagorinsky and Yoshizawa Large Eddy Simulations for a Re = 4000 plane jet simulation. Comparisons are made with Direct Numerical Simulation data and measurements. Global properties of the jet such as the spreading and centreline velocity decay rates are investigated. The mean-flow and turbulence parameters in the self-similar region are also studied. All models generally give encouraging agreement with the Direct Numerical Simulation data and reliable measurements. Solution differences for the models are relatively minor, none giving clear improvements for all data comparisons.

Mixed nonlinear LES for DES suitable flows

It is shown that a new mixed nonlinear/eddy viscosity LES model reproduces profiles better than a number of competing nonlinear and mixed models for plane channel flow. The objective is an LES method that produces a fully resolved turbulent boundary layer and could be applied to a variety of aerospace problems that are currently studied with RANS, RANS-LES, or DES methods that lack a true turbulent boundary layer. There are two components to the new model. One an eddy viscosity based upon the advected subgrid scale energy and a relatively small coefficient. Second, filtered nonlinear terms based upon the Leray regularization. Coefficients for the eddy viscosity and nonlinear terms come from LES tests in decaying, isotropic turbulence. Using these coefficients, the velocity profile matches measurements data at Reτ ≈ 1000 exactly. Profiles of the components of kinetic energy have the same shape as in the experiment, but the magnitudes differ by about 25%. None of the competing LES gets the shape correct. This method does not require extra operations at the transition between the boundary layer and the interior flow.

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity. © 2012 Trotta et al.