Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit

Structure Preserving Model Reduction of Parametric Hamiltonian Systems

While reduced-order models (ROMs) have been popular for efficiently solving large systems of differential equations, the stability of reduced models over long-time integration is of present challenges. We present a greedy approach for ROM generation of parametric Hamiltonian systems that captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis. We use the error in the Hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors. Under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters, we show that the greedy algorithm converges with exponential rate. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure. This enables the reduction of the computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy, and stability of this model reduction technique is illustrated through simulations of the parametric wave equation and the parametric Schrodinger equation

Deciphering the Firing Patterns of Hippocampal Neurons During Sharp-Wave Ripples

The hippocampus is essential for learning and memory. Neurons in the rat hippocampus selectively fire when the animal is at specific locations - place fields - within an environment. Place fields corresponding to such place cells tile the entire environment, forming a stable spatial map supporting navigation and planning. Remarkably, the same place cells reactivate together outside of their place fields and in coincidence with sharp-wave ripples (SWRs) - dominant electrical field oscillations (150-250 Hz) in the hippocampus. These offline SWR events frequently occur during quiet wake periods in the middle of exploration and the follow-up slow-wave sleep and are associated with spatial memory performance and stabilization of spatial maps. Therefore, deciphering the firing patterns during these events is essential to understanding offline memory processing.I provide two novel methods to analyze the SWRs firing patterns in this dissertation project. The first method uses hidden Markov models (HMM), in which I model the dynamics of neural activity during SWRs in terms of transitions between distinct states of neuronal ensemble activity. This method detects consistent temporal structures over many instances of SWRs and, in contrast to standard approaches, relaxes the dependence on positional data during the behavior to interpret temporal patterns during SWRs. To validate this method, I applied the method to quiet wake SWRs. In a simple spatial memory task in which the animal ran on a linear track or in an open arena, the individual states corresponded to the activation of distinct group of neurons with inter-state transitions that resembled the animal’s trajectories during the exploration. In other words, this method enabled us to identify the topology and spatial map of the explored environment by dissecting the firings occurring during the quiescence periods’ SWRs. This result indicated that downstream brain regions may rely only on SWRs to uncover hippocampal code as a substrate for memory processing. I developed a second analysis method based on the principles of Bayesian learning. This method enabled us to track the spatial tunings over the sleep following exploration of an environment by taking neurons’ place fields in the environment as the prior belief and updating it using dynamic ensemble firing patterns unfolding over time. This method introduces a neuronal-ensemble-based approach that calculates tunings to the position encoded by ensemble firings during sleep rather than the animal’s actual position during exploration. When I applied this method to several datasets, I found that during the early slow-wave sleep after an experience, but not during late hours of sleep or sleep before the exploration, the spatial tunings highly resembled the place fields on the track. Furthermore, the fidelity of the spatial tunings to the place fields predicted the place fields’ stability when the animal was re-exposed to the same environment after ~ 9h. Moreover, even for neurons with shifted place fields during re-exposure, the spatial tunings during early sleep were predictive of the place fields during the re-exposure. These results indicated that early sleep actively maintains or retunes the place fields of neurons, explaining the representational drift of place fields across multiple exposures

Honeybees solve a multi-comparison ranking task by probability matching

Honeybees forage on diverse flowers which vary in the amount and type of rewards they offer, and bees are challenged with maximizing the resources they gather for their colony. That bees are effective foragers is clear, but how bees solve this type of complex multi-choice task is unknown. Here, we set bees a five-comparison choice task in which five colours differed in their probability of offering reward and punishment. The colours were ranked such that high ranked colours were more likely to offer reward, and the ranking was unambiguous. Bees' choices in unrewarded tests matched their individual experiences of reward and punishment of each colour, indicating bees solved this test not by comparing or ranking colours but by basing their colour choices on their history of reinforcement for each colour. Computational modelling suggests a structure like the honeybee mushroom body with reinforcement-related plasticity at both input and output can be sufficient for this cognitive strategy. We discuss how probability matching enables effective choices to be made without a need to compare any stimuli directly, and the use and limitations of this simple cognitive strategy for foraging animals

Bumblebees learn a relational rule but switch to a win-stay/lose-switch heuristic after extensive training

Mapping animal performance in a behavioral task to underlying cognitive mechanisms and strategies is rarely straightforward, since a task may be solvable in more than one manner. Here, we show that bumblebees perform well on a concept-based visual discrimination task but spontaneously switch from a concept-based solution to a simpler heuristic with extended training, all while continually increasing performance. Bumblebees were trained in an arena to find rewards on displays with shapes of different sizes where they could not use low-level visual cues. One group of bees was rewarded at displays with larger shapes and another group at displays with smaller shapes. Analysis of total choices shows bees increased their performance over 30 bouts to above chance. However, analyses of first and sequential choices suggest that after approximately 20 bouts, bumblebees changed to a win-stay/lose-switch strategy. Comparing bees’ behavior to a probabilistic model based on a win-stay/lose-switch strategy further supports the idea that bees changed strategies with extensive training. Analyses of unrewarded tests indicate that bumblebees learned and retained the concept of relative size even after they had already switched to a win-stay, lost-shift strategy. We propose that the reason for this strategy switching may be due to cognitive flexibility and efficiency

Goal-oriented Uncertainty Quantification for Inverse Problems via Variational Encoder-Decoder Networks

In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient goal-oriented uncertainty quantification for inverse problems. Contrary to standard inverse problems, these approaches are \emph{goal-oriented} in that the goal is to estimate some quantities of interest (QoI) that are functions of the solution of an inverse problem, rather than the solution itself. Moreover, we are interested in computing uncertainty metrics associated with the QoI, thus utilizing a Bayesian approach for inverse problems that incorporates the prediction operator and techniques for exploring the posterior. This may be particularly challenging, especially for nonlinear, possibly unknown, operators and nonstandard prior assumptions. We harness recent advances in machine learning, i.e., VED networks, to describe a data-driven approach to large-scale inverse problems. This enables a real-time goal-oriented uncertainty quantification for the QoI. One of the advantages of our approach is that we avoid the need to solve challenging inversion problems by training a network to approximate the mapping from observations to QoI. Another main benefit is that we enable uncertainty quantification for the QoI by leveraging probability distributions in the latent space. This allows us to efficiently generate QoI samples and circumvent complicated or even unknown forward models and prediction operators. Numerical results from medical tomography reconstruction and nonlinear hydraulic tomography demonstrate the potential and broad applicability of the approach.Comment: 28 pages, 13 figure

Rail Track Detection and Projection-Based 3D Modeling from UAV Point Cloud

The expansion of the railway industry has increased the demand for the three-dimensional modeling of railway tracks. Due to the increasing development of UAV technology and its application advantages, in this research, the detection and 3D modeling of rail tracks are investigated using dense point clouds obtained from UAV images. Accordingly, a projection-based approach based on the overall direction of the rail track is proposed in order to generate a 3D model of the railway. In order to extract the railway lines, the height jump of points is evaluated in the neighborhood to select the candidate points of rail tracks. Then, using the RANSAC algorithm, line fitting on these candidate points is performed, and the final points related to the rail are identified. In the next step, the pre-specified rail piece model is fitted to the rail points through a projection-based process, and the orientation parameters of the model are determined. These parameters are later improved by fitting the Fourier curve, and finally a continuous 3D model for all of the rail tracks is created. The geometric distance of the final model from rail points is calculated in order to evaluate the modeling accuracy. Moreover, the performance of the proposed method is compared with another approach. A median distance of about 3 cm between the produced model and corresponding point cloud proves the high quality of the proposed 3D modeling algorithm in this study

Constitutionalism and Democracy Dataset, Version 1.0

The main objective of the CDD is to quantify the process of constitution-making since 1974. This is the first public release of any data on the process of constitution-making. This release includes data on 144 national constitutions promulgated in 119 countries from 1974 to 2014. The unit of analysis in the data is national constitutions. The data in this release includes only “new” constitutions and does not include suspended, re-installed, amended, or interim constitutions. In this release, only countries with a population larger than 500,000 are included. The authors intend to update the data by including all countries, expanding the time frame, and adding new variables which record the process of constitution-making

Can constitutions improve democracy? Sometimes, but not always

Does adopting a constitution lead to better democracy? Not necessarily, write Todd A. Eisenstadt, A Carl LeVan, and Tofigh Maboudi, who studied 138 constitutions from a 37 year-period. They find that in many countries, the constitution-making processes did not incorporate broad public consultation, meaning that the drafters were often able to grab power by codifying benefits for themselves and for their supporters