The Wilson-Cowan equations represent a model for the mean activities of localized excitatory and inhibitory populations in sensory cortex. In this document, we extend this model to include spatially-distributed connections in a 1D continuum model to study spatiotemporal patterns, such as traveling waves and doubly periodic patterns. We use bifurcation theory and continuation methods to understand how these organized patterns of activity arise in the network. In addition, we often simulate a (spatial) discretization of the network (to approximate the continuum) and compare these with our analytical theory to give evidence as to how these patterns may become unstable. In the later chapters, we make comparisons with a nonsmooth version of the model to understand the consequences of this approximation.
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