On the Open Convexity of Neural Codes with five Neurons
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The process through which the brain interprets and understands environmental spatial information has been a major area of interest in the field of Neuroscience. The brain, through the use of receptive fields and corresponding neurons called ``place cells'', naturally segments its environment into distinct regions of space. This process of segmenting and labeling regions can be abstracted into the concept of a Neural Code, which is a special set of labels called ``codewords'' that describe these distinct regions. Given the layout of a set of receptive fields, it is easy to generate the corresponding Neural Code. However finding a corresponding set of open convex receptive fields given a Neural Code remains an open problem. Recent research in this area has lead to the classification of all Neural Codes which have an open convex realization for codes of up to four Neurons. In this thesis, we begin a classification of codes with an open convex realization for five Neurons.