NSF Award
0908528
This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).<br/><br/>In the past several decades neuroscience and biological science have greatly benefited from insights gathered from mathematical models and studies. The investigators will study the generic behaviors of periodically forced Hopf oscillators, based on a more general normal form, in the analysis of mammalian auditory system. This includes gaining an understanding of the implication of multiple periodic responses. A very simple mode-interaction bifurcation with square symmetry that may lead to large-scale symmetric chaotic attractors will be investigated as well. Such symmetry is defined in a natural way that physical experiments can be performed more easily, such as to have a reaction-diffusion model setup more naturally. The research of planar neural field model used in developing studies of brain?s electrical activity will also be carried out. The model studied will incorporate axonal delay in nonlocal interactions, and we investigate such effects on patterning. The investigation of a simple equivariant system reveals a heteroclinic-like orbit which does not require a perturbation. This study will provide an alternative mechanism to observe such cycle without going through symmetry breaking bifurcation.<br/><br/><br/>There has been no abstract theory for observing or understanding multiple periodic responses in modeling auditory system, hence this study will be of interest in neuroscience study. The success of large-scale symmetry chaos study would be the first situation where widespread symmetric chaos could be systematically investigated in experimental situations. The study of neural field model with nonlocal connections will provide patterning resulting from the delay effect, thus help to further develop an understanding in the studies of epilepsy, coma and brain injury. Heteroclinic-like cycles play a role in analyzing neural computation, hunting behavior and olfaction selection. Graduate students involved in this research will receive training in an interdisciplinary field, and will gain a broad perspective on mathematical neuroscience. The active involvement of the investigators in student research and course development will provide an opportunity to translate the research here into compelling educational topics.
