Neural fields models have been developed. to emulate large scale brain dynamics. They exhibit similar types of patterns as observed in real cortical tissue, such as travelling waves and persistent localised activity. The study of neural field models is yet a growing field of research, and in this thesis we contribute by developing new approaches to the analysis of pattern formation. A particular focus is on interface methods in one and two spatial dimensions. In the first part of this thesis we Study the influence of inhomogeneities on the velocity of propagating waves. We examine periodically modulated connectivity functions as well as fluctuating firing thresholds. For strong inhomogeneities we observe wave propagation failure and the emergence of stable localised solutions that do not exist in the homogeneous model. In the second part we develop a method to approximate stationary localised solutions and travelling waves in neural field models with sigmoidal firing rates. In particular, we devise a scheme that approximates the slope of these solutions and yields refined results upon iteration. We calculate explicit solutions for piecewise linear and piecewise polynomial firing rates. In the third part we develop an interface approach for planar neural field models. We derive the equations of motion for a certain class of synaptic connectivity function. In the interface description the evolution of a contour, which is defined by a level set condition, is governed by the normal velocity which depends exclusively on the shape of the contour. We present results for the existence and stability of various types of patterns. The interface description is also incorporated into a numerical scheme which allows to investigate pattern formation beyond instabilities.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:597110 |
| Date | January 2012 |
| Creators | Schmidt, Helmut |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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