CAS: Mathematics & Statistics: Scholarly Works
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Item Catalyst: fast biochemical modeling with Julia(2022-08) Loman, Torkel E.; Ma, Yingbo; Ilin, Vasily; Gowda, Shashi; Korsbo, Niklas; Yewale, Nikhil; Rackauckas, Chris; Isaacson, Samuel A.Item Fluctuation analysis for particle-based stochastic reaction-diffusion models(2022-06) Heldman, Max; Isaacson, Samuel; Ma, Jingwei; Spiliopoulos, KonstantinosRecent works have derived and proven the large-population mean-field limit for several classes of particle-based stochastic reaction-diffusion (PBSRD) models. These limits correspond to systems of partial integral-differential equations (PIDEs) that generalize standard mass-action reaction-diffusion PDE models. In this work we derive and prove the next order fluctuation corrections to such limits, which we show satisfy systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how including the fluctuation corrections can enable the accurate estimation of higher order statistics of the underlying PBSRD model.Item How reaction-diffusion PDEs approximate the large-population limit of stochastic particle models(Society for Industrial & Applied Mathematics (SIAM), 2021-01) Isaacson, Samuel A.; Ma, Jingwei; Spiliopoulos, KonstantinosReaction-diffusion PDEs and particle-based stochastic reaction-diffusion (PBSRD) models are commonly used approaches for modeling the spatial dynamics of chemical and biological systems. Standard reaction-diffusion PDE models ignore the underlying stochasticity of spatial transport and reactions and are often described as appropriate in regimes where there are large numbers of particles in a system. Recent studies have proven the rigorous large-population limit of PBSRD models, showing the resulting mean-field models (MFMs) correspond to nonlocal systems of partial-integro differential equations. In this work we explore the rigorous relationship between standard reaction-diffusion PDE models and the derived MFM. We prove that the former can be interpreted as an asymptotic approximation to the later in the limit that bimolecular reaction kernels are short-range and averaging. As the reactive interaction length scale approaches zero, we prove the MFMs converge at second order to standard reaction-diffusion PDE models. In proving this result we also establish local well-posedness of the MFM model in time for general systems and global well-posedness for specific reaction systems and kernels. Finally, we illustrate the agreement and disagreement between the MFM, SM, and underlying particle model for several numerical examples.Item A tale of three curves(2022-10-27) Balakrishnan, JenniferIn this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.Item Rapid synaptic and gamma rhythm signature of mouse critical period plasticity(2023) Kopell, Nancy; Quast, K.B.; Reh, R.K.; Caiati, M.D.; McCarthy, M.M.; Hensch, T.K.Item Perturbation detection through modeling of gene expression on a latent biological pathway network: a Bayesian hierarchical approach(Taylor & Francis, 2016) Pham, Lisa M.; Carvalho, Luis; Schaus, Scott; Kolaczyk, Eric D.Cellular response to a perturbation is the result of a dynamic system of biological variables linked in a complex network. A major challenge in drug and disease studies is identifying the key factors of a biological network that are essential in determining the cell's fate. Here our goal is the identification of perturbed pathways from high-throughput gene expression data. We develop a three-level hierarchical model, where (i) the first level captures the relationship between gene expression and biological pathways using confirmatory factor analysis, (ii) the second level models the behavior within an underlying network of pathways induced by an unknown perturbation using a conditional autoregressive model, and (iii) the third level is a spike-and-slab prior on the perturbations. We then identify perturbations through posterior-based variable selection. We illustrate our approach using gene transcription drug perturbation profiles from the DREAM7 drug sensitivity predication challenge data set. Our proposed method identified regulatory pathways that are known to play a causative role and that were not readily resolved using gene set enrichment analysis or exploratory factor models. Simulation results are presented assessing the performance of this model relative to a network-free variant and its robustness to inaccuracies in biological databases.Item Calibrated model criticism using split predictive checks(2022-03-29) Li, Jiawei; Huggins, JonathanChecking how well a fitted model explains the data is one of the most fundamental parts of a Bayesian data analysis. However, existing model checking methods suffer from trade-offs between being well-calibrated, automated, and computationally efficient. To overcome these limitations, we propose split predictive checks (SPCs), which combine the ease-of-use and speed of posterior predictive checks with the good calibration properties of predictive checks that rely on model-specific derivations or inference schemes. We develop an asymptotic theory for two types of SPCs: single SPCs and the divided SPC. Our results demonstrate that they offer complementary strengths: single SPCs provide superior power in the small-data regime or when the misspecification is significant and divided SPCs provide superior power as the dataset size increases or when the form of misspecification is more subtle. We validate the finite-sample utility of SPCs through extensive simulation experiments in exponential family and hierarchical models, and provide four real-data examples where SPCs offer novel insights and additional flexibility beyond what is available when using posterior predictive checks.Item Robust, automated, and accurate black-box variational inference(2022-03-29) Welandawe, Manushi; Riis Anderson, Michael; Vehtari, Aki; Huggins, JonathanBlack-box variational inference (BBVI) now sees widespread use in machine learning and statistics as a fast yet flexible alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, stochastic optimization methods for BBVI remain unreliable and require substantial expertise and hand-tuning to apply effectively. In this paper, we propose Robust, Automated, and Accurate BBVI (RAABBVI), a framework for reliable BBVI optimization. RAABBVI is based on rigorously justified automation techniques, includes just a small number of intuitive tuning parameters, and detects inaccurate estimates of the optimal variational approximation. RAABBVI adaptively decreases the learning rate by detecting convergence of the fixed--learning-rate iterates, then estimates the symmetrized Kullback--Leiber (KL) divergence between the current variational approximation and the optimal one. It also employs a novel optimization termination criterion that enables the user to balance desired accuracy against computational cost by comparing (i) the predicted relative decrease in the symmetrized KL divergence if a smaller learning were used and (ii) the predicted computation required to converge with the smaller learning rate. We validate the robustness and accuracy of RAABBVI through carefully designed simulation studies and on a diverse set of real-world model and data examples.Item Broadband slow-wave modulation in posterior and anterior cortex tracks distinct states of propofol-induced unconsciousness(Springer Science and Business Media LLC, 2020-08-13) Stephen, Emily P.; Hotan, Gladia C.; Pierce, Eric T.; Harrell, P. Grace; Walsh, John L.; Brown, Emery N.; Purdon, Patrick L.A controversy has developed in recent years over the roles of frontal and posterior cortices in mediating consciousness and unconsciousness. Disruption of posterior cortex during sleep appears to suppress the contents of dreaming, yet activation of frontal cortex appears necessary for perception and can reverse unconsciousness under anesthesia. We used anesthesia to study how regional cortical disruption, mediated by slow wave modulation of broadband activity, changes during unconsciousness in humans. We found that broadband slow-wave modulation enveloped posterior cortex when subjects initially became unconscious, but later encompassed both frontal and posterior cortex when subjects were more deeply anesthetized and likely unarousable. Our results suggest that unconsciousness under anesthesia comprises several distinct shifts in brain state that disrupt the contents of consciousness distinct from arousal and awareness of those contents.Item State space methods for phase amplitude coupling analysis(Springer Science and Business Media LLC, 2022-09-24) Soulat, Hugo; Stephen, Emily P.; Beck, Amanda M.; Purdon, Patrick L.Phase amplitude coupling (PAC) is thought to play a fundamental role in the dynamic coordination of brain circuits and systems. There are however growing concerns that existing methods for PAC analysis are prone to error and misinterpretation. Improper frequency band selection can render true PAC undetectable, while non-linearities or abrupt changes in the signal can produce spurious PAC. Current methods require large amounts of data and lack formal statistical inference tools. We describe here a novel approach for PAC analysis that substantially addresses these problems. We use a state space model to estimate the component oscillations, avoiding problems with frequency band selection, nonlinearities, and sharp signal transitions. We represent cross-frequency coupling in parametric and time-varying forms to further improve statistical efficiency and estimate the posterior distribution of the coupling parameters to derive their credible intervals. We demonstrate the method using simulated data, rat local field potentials (LFP) data, and human EEG data.Item Sevoflurane induces coherent slow-delta oscillations in rats(Frontiers Media SA, 2017) Guidera, Jennifer A.; Taylor, Norman E.; Lee, Justin T.; Vlasov, Ksenia Y.; Pei, JunZhu; Stephen, Emily P.; Mayo, J. Patrick; Brown, Emery N.; Solt, KenAlthough general anesthetics are routinely administered to surgical patients to induce loss of consciousness, the mechanisms underlying anesthetic-induced unconsciousness are not fully understood. In rats, we characterized changes in the extradural EEG and intracranial local field potentials (LFPs) within the prefrontal cortex (PFC), parietal cortex (PC), and central thalamus (CT) in response to progressively higher doses of the inhaled anesthetic sevoflurane. During induction with a low dose of sevoflurane, beta/low gamma (12-40 Hz) power increased in the frontal EEG and PFC, PC and CT LFPs, and PFC-CT and PFC-PFC LFP beta/low gamma coherence increased. Loss of movement (LOM) coincided with an abrupt decrease in beta/low gamma PFC-CT LFP coherence. Following LOM, cortically coherent slow-delta (0.1-4 Hz) oscillations were observed in the frontal EEG and PFC, PC and CT LFPs. At higher doses of sevoflurane sufficient to induce loss of the righting reflex, coherent slow-delta oscillations were dominant in the frontal EEG and PFC, PC and CT LFPs. Dynamics similar to those observed during induction were observed as animals emerged from sevoflurane anesthesia. We conclude that the rat is a useful animal model for sevoflurane-induced EEG oscillations in humans, and that coherent slow-delta oscillations are a correlate of sevoflurane-induced behavioral arrest and loss of righting in rats.Item Global solutions to the stochastic reaction-diffusion equation with superlinear accretive reaction term and superlinear multiplicative noise term on a bounded spatial domain(American Mathematical Society (AMS), 2022-09-02) Salins, MichaelWe describe sufficient conditions on the reaction terms and multiplicative noise terms of a stochastic reaction-diffusion equation that guarantee that the solutions never explode. Both the reaction term and multiplicative noise terms are allowed to grow superlinearly.Item Noncommutative geometry of computational models and uniformization for framed quiver varietiesLau, Siu; Jeffreys, GeorgeItem Quantum finite automata and quiver algebras(MDPI, 2022-12-14) Lau, SiuItem Mirror symmetry for quiver algebroid stacksLau, Siu; Tan, Ju; Junzheng, NanIn this paper, we construct noncommutative algebroid stacks and the associated mirror functors for a symplectic manifold. First, we formulate a version of stack that is well adapted for gluing quiver algebras with different numbers of vertices. Second, we develop a representation theory of A∞ categories by quiver stacks. A key step is constructing an extension of the A∞ category over a quiver stack of a collection of nc-deformed objects. The extension involves non-trivial gerbe terms, which play an important role for quiver algebroid stacks. Third, we apply the theory to construct mirror quiver stacks of local Calabi-Yau manifolds. In this paper, we focus on nc local projective plane. This example has a compact divisor which gives rise to interesting monodromy and homotopy terms which can be found from mirror symmetry. Geometrically, we find a new method of mirror construction by gluing with a middle agent using Floer theory. The method makes crucial use of the extension of Fukaya category over quiver stacks.Item Recent progress on SYZ mirror symmetry for some non-compact Calabi-Yau surfacesLin, Yu-Shen; Collins, TristanItem Period domains of gravitational instantonsLin, Yu-ShenItem SYZ mirror symmetry for del Pezzo surfaces and affine structuresLin, Yu-Shen; Lau, Siu-Cheong; Lee, Tsung-JuItem Normalization effects on deep neural networks(2022-09-02) Spiliopoulos, Konstantinos; Yu, JiahuiItem Moderate deviations for fully-coupled multiscale weakly interacting particle systems(2022-02-17) Spiliopoulos, Konstantinos; Bezemek, ZacharyWe consider a collection of fully coupled weakly interacting diffusion processes moving in a two-scale environment. We study the moderate deviations principle of the empirical distribution of the particles’ positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods, which provide a convenient representation for the moderate deviations rate function in terms of an effective mean field control problem. We rigorously obtain equivalent representations for the moderate deviations rate function in an appropriate “negative Sobolev” form, which is reminiscent of the large deviations rate function for the empirical measure of weakly interacting diffusions obtained in the 1987 seminal paper by Dawson-G¨artner. In the course of the proof we obtain related ergodic theorems and we rigorously study the regularity of Poisson type of equations associated to McKean-Vlasov problems, both of which are topics of independent interest.