Differential and Integral Equations in Neurobiology

用非线性动力学的方法研究了模型神经元网络中振子与非振子之间的耦合问题。具体地说，通过将复杂的基于电导的模型简化为标量空间模型，平均法被用来分析空间连接网络中的波的终止。将相平面方法应用于约化系统。在单尖峰级别也使用平均，以了解具有不同类型连接的耦合网络中同步和异步之间的转换。人口密度方法和线性化稳定性分析将区分神经振荡器空间分布网络中同步波、群波和传播波的开始。可兴奋细胞和振荡细胞之间的间接耦合将利用相位重置曲线和可兴奋电缆的色散特性相结合来分析。当神经元连接在一起时，它们通常可以产生持续的活动。这种持续的活动被认为与短期记忆有关--即动物或人类如何“持有一种思想”。什么样的互动破坏了这一点，哪些是维持这种活动所必需的，这是本提案中提出的一些问题。当活动持续太久时，就会出现某些病理现象，如癫痫。因此，这项提议的一个目标是了解如何在产生稳定的持续活动的能力和防止其病理性传播到安静地区之间取得平衡。当神经元放电时，它们与其他神经元进行交流，根据相互作用的不同，结果可能是神经元想要一起放电，或者它们想要异步放电。后者对于持久化活动非常有用。另一方面，同步对于几个正常的生理过程是至关重要的。例如，众所周知，大脑底部的某些细胞组织荷尔蒙的输出。这些细胞的电活动是同步的，但这种同步的机制仍不清楚，因为各个振荡神经元之间没有直接联系。我们将研究间接耦合产生同步行为的机制。这项建议中开发的工具和方法的应用将远远超出神经科学，因为从单细胞水平到种群之间的生态相互作用，“信息”的同步和传播问题在生物学中无处不在。

Methods from nonlinear dynamics are used to study the coupling between oscillators and non-oscillators in model neuronal networks. Specifically, averaging is used to analyze the termination of waves in spatially connected networks by reducing the complicated conductance-based models to scalar spatial models. Phase-plane methods are applied to the reduced systems. Averaging is also used at the single spike level to understand the transition between synchrony and asynchrony in coupled networks that have different types of connections. Population density methods and linearized stability analysis will distinguish between the onset of synchrony, clusters, and propagating waves in spatially distributed networks of neural oscillators. Indirect coupling between excitable and oscillatory cells will be analyzed using a combination of phase-resetting curves and the dispersive properties of excitable cables.When neurons are connected together, they can often produce persistent activity. Such persistent activity has been implicated in short-term memory -- how an animal or human "holds a thought." What kinds of interactions disrupt this and which are necessary to maintain the activity are some of the questions that are asked in this proposal. When activity is too persistent then certain pathologies arise such as epilepsy. Thus, one goal of this proposal is to understand how to strike a balance between the ability to produce stable persistent activity while preventing its pathological propagation into quiet regions. When neurons fire they communicate with other neurons and depending on the interactions, the result can be that the neurons want to fire together or they want to fire asynchronously. The latter is useful for persistent activity. Synchrony on the other hand is crucial for several normal physiological processes. For example, it is known that certain cells in the base of the brain organize the output of hormones. The electrical activity of these cells is synchronized yet the mechanisms for this synchrony remain unknown since there are no direct connections between the individual oscillating neurons. We will study mechanism through which indirect coupling can produce synchronous behavior. Tools and methods developed in this proposal will have applications well beyond neuroscience, as the questions of synchrony and propagation of "information" are ubiquitous in biology from the single cell level on up to the ecological interactions between populations.