Continuum neural field models mimic the large scale spatio-temporal dynamics of interacting neurons on a cortical surface. For smooth Mexican hat kernels, with short-range excitation and long-range inhibition, they support various localised structures as well as travelling waves similar to those seen in real cortex. These non-local models have been extensively studied, both analytically and numerically, yet there remain open challenges in their study. Here we provide new numerical and analytical treatments for the study of spatio-temporal pattern formation in neural field models. In this context, the description of spreading patterns with a well identified interface is of particular interest, as is their dependence on boundary conditions. This Thesis is dedicated to the analyses of one- and two-dimensional localised states as well as travelling waves in neural fields. Firstly we analyse the effects of Dirichlet boundary conditions on shaping and creating localised bumps in one- dimensional spatial models, and then on the development of labyrinthine structures in two spatial dimensions. Linear stability analysis is used to understand how spatially extended patterns may develop in the absence and presence of boundary conditions. For the case without boundary conditions we recover the results of Amari, namely the widest bump among two branches of solutions is stable. However, new stable solutions can arise with an imposed Dirichlet boundary condition. For a Heaviside non-linearity, the Amari model allows a description of solutions using an equivalent interface dynamics. We generalise this reduced, yet exact, description by deriving a normal velocity rule that can account for boundary conditions. We extend this approach to further treat neural field models with spike frequency adaptation. These can exhibit breathers and travelling waves. The latter can take the form of spiral waves, to which we devote particular attention. We further study neural fields on feature spaces in the primary visual cortex (V$1$), where cells respond preferentially to edges of a particular orientation. Considering a general form of the synaptic kernel which includes an orientation preference at each spatial point, we present the construction and stability of orientation bumps, as well as stripes. To date there has been surprisingly little analysis of spatio-temporal pattern formation in neural field equations described on curved surfaces. Finally, we study travelling fronts and pulses on non-flat geometries, where we consider the effects of inhomogeneities on the propagation velocity of these waves. In all sections, theoretical results for pattern formation are shown to be in excellent agreement with simulations of the full neural field models.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:734396 |
| Date | January 2017 |
| Creators | Gokce, Aytul |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://eprints.nottingham.ac.uk/48185/ |
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