Spatiotemporal Dynamics of Nonlocally Connected Networks

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The perception of the world by the brain is governed both by the external stimuli that it receives, by its constant ongoing activity, and by its prior experience. The ongoing activity is formed by the interaction of the millions of cells that comprise the brain and that sit in a delicately balanced state in order to respond quickly and sensitively to the rapidly changing world. These cells do not form random patterns, but rather produce patterned activity such as waves and rhythms that can form the precursors to complex behaviors such as speaking and walking. This ongoing and evoked behavior is carefully controlled by a balance of positive (excitatory) and negative (inhibitory) influences. The loss of this balance can disrupt normal behavior and lead to diseases such as epilepsy and schizophrenia. This research project uses mathematical models of the nervous system to provide a way to test hypotheses about brain activity put forth by experimentalists and to suggest new experiments based on the predictions of these models. In addition the project will also provide research training opportunities for graduate students. Ongoing activity in the nervous system and how it impacts sensory and other inputs is the subject of much recent experimental activity. In particular, it is clear that the intrinsic interactions between neuronal circuits in absence of inputs can have a strong impact on how the system responds to incoming stimuli even at the large-scale cognitive level. This project will apply nonlinear dynamics methods to problems in theoretical neuroscience dealing with this question. Various forms of spatiotemporal activity, including spatially localized activity, oscillations, and propagating waves, are observed experimentally. There is now a good deal of evidence that the oscillatory activity that is ubiquitously found in the brain is not synchronized, but, rather, takes the forms of propagating waves and other types of spatiotemporal dynamics. The research will use mathematical models to study neurons that are connected to each other through nonlocal, spatially extended connections. The models of these networks involve nonlocal interactions leading to integro-differential equations. The project aims to develop new techniques for the analysis of such nonlocal equations, including novel perturbation methods, dimension reduction methods, and techniques for localizing the operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.