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How the brain preserves information about more than one stimulus at a time remains poorly understood. We recently showed that when more than one stimulus is present, single neurons may fluctuate between coding one vs. the other(s) across some time period. A critical unanswered question is whether and how any such fluctuations are coordinated across the neural population. Here, we analyzed the spike count (“noise”) correlations observed between pairs of primary visual cortex (V1) neurons under a variety of conditions. We report that when two separate grating stimuli are presented simultaneously, distinct distributions of positive and negative correlations emerge, depending on whether the two neurons in the pair both respond more strongly to the same vs. different individual stimuli. Neural pairs that shared the same stimulus preference were more likely to show positively correlated spike count variability whereas those with different preferences were more likely to show negative correlations, suggesting that the population response to one particular stimulus may be enhanced over the other on any given trial. This pattern of results was not present when the two gratings were superimposed and formed a single plaid, supporting the interpretation that the pattern of correlated fluctuations is related to the segregation of individual objects in the visual scene.

We introduce a Bayesian semiparametric methodology for joint quantile regression with linearity and piecewise linearity constraints. We develop a probability model for all quantile curves in a continuum that define a coherent sampling distribution of the response variable. We provide a detailed illustration of model fitting and inference by analyzing wind speed trends of tropical cyclones in the North Atlantic.

This thesis naturally divides itself into two sections. The first two chapters concernthe development of Bayesian semi-parametric models for arrival times. Chapter 2considers Bayesian inference for a Gaussian process modulated temporal inhomogeneous Poisson point process, made challenging by an intractable likelihood. The intractable likelihood is circumvented by two novel data augmentation strategies which result in Gaussian measurements of the Gaussian process, connecting the model with a larger literature on modelling time-dependent functions from Bayesian non-parametric regression to time series. A scalable state-space representation of the Matern Gaussian process in 1 dimension is used to provide access to linear time filtering algorithms for performing inference. An MCMC algorithm based on Gibbs sampling with slice-sampling steps is provided and illustrated on simulated and real datasets. The MCMC algorithm exhibits excellent mixing and scalability.Chapter 3 builds on the previous model to detect specific signals in temporal point patterns arising in neuroscience. The firing of a neuron over time in response to an external stimulus generates a temporal point pattern or ``spike train''. Of special interest is how neurons encode information from dual simultaneous external stimuli. Among many hypotheses is the presence multiplexing - interleaving periods of firing as it would for each individual stimulus in isolation. Statistical models are developed to quantify evidence for a variety of experimental hypotheses. Each experimental hypothesis translates to a particular form of intensity function for the dual stimuli trials. The dual stimuli intensity is modelled as a dynamic superposition of single stimulus intensities, defined by a time-dependent weight function that is modelled non-parametrically as a transformed Gaussian process. Experiments on simulated data demonstrate that the model is able to learn the weight function very well, but other model parameters which have meaningful physical interpretations less well.Chapters 4 and 5 concern mathematical optimization and theoretical properties of Bayesian models for variable selection. Such optimizations are challenging due to non-convexity, non-smoothness and discontinuity of the objective. Chapter 4 presents advances in continuous optimization algorithms based on relating mathematical and statistical approaches defined in connection with several iterative algorithms for penalized linearregression. I demonstrate the equivalence of parameter mappings using EM underseveral data augmentation strategies - location-mixture representations, orthogonal data augmentation and LQ design matrix decompositions. I show that thesemodel-based approaches are equivalent to algorithmic derivation via proximalgradient methods. This provides new perspectives on model-based and algorithmicapproaches, connects across several research themes in optimization and statistics,and provides access, beyond EM, to relevant theory from the proximal gradientand convex analysis literatures.Chapter 5 presents a modern and technologically up-to-date approach to discrete optimization for variable selection models through their formulation as mixed integer programming models. Mixed integer quadratic and quadratically constrained programs are developed for the point-mass-Laplace and g-prior. Combined with warm-starts and optimality-based bounds tightening procedures provided by the heuristics of the previous chapter, the MIQP model developed for the point-mass-Laplace prior converges to global optimality in a matter of seconds for moderately sized real datasets. The obtained estimator is demonstrated to possess superior predictive performance over that obtained by cross-validated lasso in a number of real datasets. The MIQCP model for the g-prior struggles to match the performance of the former and highlights the fact that the performance of the mixed integer solver depends critically on the ability of the prior to rapidly concentrate posterior mass on good models.

Current frontiers in complex stochastic modeling of high-dimensional processes include major emphases on so-called functional data: problems in which the data are snapshots of curves and surfaces representing fundamentally important scientific quantities. This thesis explores new Bayesian methodologies for functional data analysis. The first part of the thesis places emphasis on the role of factor models in functional data analysis. Data reduction becomes mandatory when dealing with such high-dimensional data, more so when data are available on a large number of individuals. In Chapter 2 we present a novel Bayesian framework which employs a latent factor construction to represent each variable by a low dimensional summary. Further, we explore the important issue of modeling and analyzing the relationship of functional data with other covariate and outcome variables simultaneously measured on the same subjects.The second part of the thesis is concerned with the analysis of circadian data. The focus is on the identification of circadian genes that is, genes whose expression levels appear to be rhythmic through time with a period of approximately 24 hours. While addressing this goal, most of the current literature does not account for the potential dependence across genes. In Chapter 4, we propose a Bayesian approach which employs latent factors to accommodate dependence and verify patterns and relationships between genes, while representing the true gene expression trajectories in the Fourier domain allows for inference on period, phase, and amplitude of the signal.The third part of the thesis is concerned with the statistical analysis of computer models (simulators). The heavy computational demand of these input-output maps calls for statistical techniques that quickly estimate the surface output at untried inputs given a few preliminary runs of the simulator at a set design points. In this regard, we propose a Bayesian methodology based on a non-stationary Gaussian process. Relying on a model-based assessment of uncertainty, we envision a sequential design technique which helps choosing input points where the simulator should be run to minimize the uncertainty in posterior surface estimation in an optimal way. The proposed non-stationary approach adapts well to output surfaces of unconstrained shape.

How neural representations preserve information about multiple stimuli is mysterious. Because tuning of individual neurons is coarse (for example, visual receptive field diameters can exceed perceptual resolution), the populations of neurons potentially responsive to each individual stimulus can overlap, raising the question of how information about each item might be segregated and preserved in the population. We recently reported evidence for a potential solution to this problem: when two stimuli were present, some neurons in the macaque visual cortical areas V1 and V4 exhibited fluctuating firing patterns, as if they responded to only one individual stimulus at a time. However, whether such an information encoding strategy is ubiquitous in the visual pathway and thus could constitute a general phenomenon remains unknown. Here we provide new evidence that such fluctuating activity is also evoked by multiple stimuli in visual areas responsible for processing visual motion (middle temporal visual area, MT), and faces (middle fundus and anterolateral face patches in inferotemporal cortex - areas MF and AL), thus extending the scope of circumstances in which fluctuating activity is observed. Furthermore, consistent with our previous results in the early visual area V1, MT exhibits fluctuations between the representations of two stimuli when these form distinguishable objects but not when they fuse into one perceived object, suggesting that fluctuating activity patterns may underlie visual object formation. Taken together, these findings point toward an updated model of how the brain preserves sensory information about multiple stimuli for subsequent processing and behavioral action., Impact Statement: We find neural fluctuations in multiple areas along the visual cortical hierarchy that could allow the brain to represent distinct co-occurring visual stimuli.