Dynamic square patterns in two dimensional neural fields

Abstract

The goal of this thesis is to study the emergence of spatiotemporal waves in neural field models. Neural field models aim to describe the activity of populations of neurons at a mesoscopic scale, considering averaged neuronal states dependent on continuous space and time. Mathematically, they are composed of spatial and temporal integral operators on domains of anatomical interest. The cortex is modelled as a two dimensional sheet, and under physiological assumptions for the spatial extent of connectivities, it is shown when the principal transition from resting to active states will result in the formation of waves. This thesis starts with a derivation for the integral operators from a physiological viewpoint. The notion of a dynamical system is then introduced, and theory relevant to the spontaneous emergence of activity is discussed. The thesis progresses to applying the dynamical systems view to neural fields, leading to an understanding of the transitions from inactive resting states to space dependent temporal oscillations - waves. For tractable analysis, the active states are restricted to have square periodic symmetry.