Spatiotemporal Dynamics of Nonlocally Connected Networks

大脑对世界的感知既受它所接收的外部刺激的支配，也受它持续不断的活动和它先前的经验的支配。持续的活动是由构成大脑的数百万个细胞相互作用形成的，这些细胞处于微妙的平衡状态，以便对快速变化的世界做出快速灵敏的反应。这些细胞不会形成随机的模式，而是产生模式化的活动，如波和节奏，可以形成复杂行为的前兆，如说话和行走。 这种持续的和诱发的行为受到积极（兴奋）和消极（抑制）影响的平衡的仔细控制。这种平衡的丧失会扰乱正常行为，导致癫痫和精神分裂症等疾病。该研究项目使用神经系统的数学模型来提供一种方法来测试实验学家提出的有关大脑活动的假设，并根据这些模型的预测建议新的实验。此外，该项目还将为研究生提供研究培训机会。神经系统中正在进行的活动以及它如何影响感觉和其他输入是最近许多实验活动的主题。特别是，很明显，在没有输入的情况下，神经元回路之间的内在相互作用可以对系统如何响应传入的刺激产生强烈的影响，即使在大规模的认知水平上。本计画将应用非线性动力学的方法来解决神经科学理论中的问题。各种形式的时空活动，包括空间局部活动，振荡和传播波，实验观察。现在有大量的证据表明，在大脑中普遍存在的振荡活动并不是同步的，而是以传播波和其他类型的时空动力学的形式出现的。这项研究将使用数学模型来研究通过非局部的、空间扩展的连接相互连接的神经元。这些网络的模型涉及非局部相互作用，导致积分微分方程。该项目旨在开发新的技术，分析这种非局部方程，包括新的扰动方法，降维方法，和技术本地化的operators.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.