| Description |
In this dissertation we mathematically analyze neural field models for populations of orientation-selective neurons in the primary visual cortex V1. Neural fields are nonlinear integro-differential equations approximating the average firing rate activity of populations of cortical neurons. We specifically focus on models of visual neurons that are sensitive to the orientation of local edges in a stimulus. We aim to develop mathematical methods for analyzing solutions to these equations in order to understand how such neurons process information from a visual scene. The enclosed work is a collection of independent, but related, studies using various mathematical tools to tackle these neural field equations from different directions in attempt to understand the dynamics of orientation-selective neurons. First, we use perturbation theory to derive a reduced equation approximating the spatial distribution of orientation tuning curves, which represent the encoded orientation of edges, for a special type of visual image. Next, we study spontaneous pattern formation in a one dimensional model by using bifurcation theory to analyze instabilities of nontrivial steady state solutions. Moving on, we explore a two-layer neural field model consisting of an orientation-independent layer and an orientation-dependent layer. We analyze and numerically simulate the model equations to understand how traveling waves in the former layer can drive waves in the latter, and what kind of patterns of orientation selectivity occur. Finally, we use symmetric bifurcation theory in a two dimensional model to study how the orientation-dependent circuitry can destabilize an orientation-independent solution to form stable spatio-oriented solutions. |
