Probabilistic Models of Phase Variables for Visual Representation and Neural Dynamics

My work seeks to contribute to three broad goals: predicting the computational representations found in the brain, developing algorithms that help us infer the computations that the brain performs, and producing better statistical models of natural signals. At first glance these goals may not seem compatible; however, my work finds a common thread among them through the probabilistic modeling of phase variables. My thesis is broken down into three major chapters that reflect these three goals. Within each chapter I develop novel probabilistic models of phase variables and apply these models to the invariant representation of visual motion, to the inference of connectivity in networks of coupled neural oscillators, and to the development of statistical models of edge structure in images. First, I develop a hierarchical model of visual processing that learns from the natural world the higher-order structure of visual motion by modeling phase transformations. The model exhibits an important invariance: the model represents the way the world moves irrespective of the way it looks. This model has implications for our interpretation of biological visual processing and provides a functional roll for feedback in cortex. Second, I present a model and estimation technique that captures the dynamics of coupled oscillator systems and recovers the interactions of the oscillators from measurements. From a statistical perspective, the model is the multivariate phase distribution analogue to the multivariate Gaussian distribution and the estimation technique is then analogous to finding the inverse covariance matrix for a Gaussian distribution. From a dynamical systems perspective, the technique provides a solution to the inverse problem of the generalized Kuramoto model and infers from measurements the true connectivity between oscillators even when phase correlations or other phase measurements would lead to false conclusions. This technique can be broadly applied to a range of neurobiological phenomena including the inference of cortical dynamic functional networks from phase measurements. Third, I present a model that captures aspects of the local phase structure of edges in images. We first explore the pairwise phase statistics of local, oriented filters in response to natural images and determine that pairwise phase relationships do not explain the `interesting' relationships in natural images, such as long range phase alignments. Given this finding we develop a conditional latent variable model that captures the non-stationary phase structure produced by continuous edges. This model is capable of generating long range, continuous edge structure, a hallmark of natural images. The major contributions of this thesis can be divided into two types. First, this work provides demonstrative examples of how multivariate phase distributions may be modeled in a probabilistic framework. My hope is that the models I have developed will provide the basis for additional exploration of the mathematical development of probabilistic models of phase. Second, the results obtained from applying these models have important implications for understanding invariant visual representations of motion, investigating coherence mediated intracortical communication, and describing the statistical structure of edges in natural images.