Nonlinear Differential Equations & Linear Equations in Physiology & Cell Biology.

研究者继续研究非线性动力学在细胞生物学和生理学中的应用。他研究了树突、轴突和突触延迟对耦合神经振荡器行为的影响。他开发并分析了噪声存在下的神经振荡器模型。他提出并分析了神经组织中突触波传播的多种模型，包括：1。网状核中纺锤波模型的系统数值探索。在以前的拨款中开发的软件能够交互式地求解这些积分微分方程。2. 突触诱导双稳定性分析；突触时间常数和突触强度如何影响一组突触维持两种行为模式：静息和振荡的能力？将突触耦合的生物物理模型正式简化为易于数学分析的简化模型。突触电流和固有慢电流都包含在这些模型中。4. 对上述简化模型进行严格和形式化的数学分析，特别注意保持与完整方程的定量相似性。他与Jose Gonzalez-Fernandez博士合作扩展了他的血管舒缩模型，将空间方面和通过组织床的耦合纳入其中。目的是解释克拉夫关于毛细血管募集的经典结果，并了解微血管水平上血流的动态调节。他和冈萨雷斯-费尔南德斯打算将这些结果与西格尔最近的一些数据以及以前的研究进行比较。作为附带的好处，他们希望开发一些新的方法来分析通过空间连续但非振荡介质耦合的振子系统。他继续与George Oster合作，利用线性分子马达和旋转马达的先前结果来开发和分析(i)噬菌体的门户蛋白，（ii）质子驱动马达，（iii）质子泵的模型。他与Leah edelstein - kesheet合作建立了肌动蛋白丝长度分布及其在细胞内空间变化的模型。这项工作试图运用数学方法来帮助理解生理节律。许多人类生物过程是由不同数量的消长所控制的。例如，心脏起搏器的电压，睡眠时大脑在不同状态下的规律漂移，以及运动时小动脉有节奏的收缩对肌肉中氧气流量的控制。这项工作的方法是使用计算机和数学模型。作为这项工作的一部分，开发了计算工具和可视化模拟工具。主要目标是利用这些模型来弥合细胞微观运作的细节和它们对行为的影响之间的差距。例如，当肌肉剧烈运动时，它们会消耗氧气，因此需要更多来自动脉的氧气。在形成动脉的细胞中有特殊的门，可以感知“燃料”的数量，当燃料减少时，门就会告诉动脉打开。这种反馈回路提供了一种动态改变动脉节律的方法，以确保组织得到均匀的氧合。因此，从这些通道或通道的微观层面出发，建立了一个模型，可以解释组织中氧气的消耗是如何调节的。另一个例子发生在神经系统。来自大脑不同部位的信息通过神经细胞之间的特殊连接传递，这种连接被称为突触。反应时间和认知能力取决于这种传播的速度，以及防止传播到大脑的其他区域。关于单个神经细胞对刺激的反应，我们知道得很多。模型提供了一种将许多这样的细胞耦合在一起的方法，以便了解单个细胞特性的行为后果。整体大于部分之和吗？这个提议的目的是试图回答这个问题的各种节律控制的生理过程。

Ermentrout 9626728 The investigator continues his work on the applications of nonlinear dynamics to cell biology and physiology. He studies the effects of dendritic, axonal, and synaptic delays on the behavior of coupled neural oscillators. He develops and analyzes models of neural oscillators in the presence of noise. He proposes and analyzes a variety of models for synaptic wave propagation in neural tissue including: 1. Systematic numerical exploration of models for spindle waves in the nucleus reticularis. Software developed in the previous grant is capable of interactively solving these integrodifferential equations. 2. Analysis of synaptically induced bistability; how do synaptic time constants and synaptic strengths effect the ability of a group of synapses to maintain two modes of behavior: resting and oscillatory 3. Formal reduction of synaptically coupled biophysical models to simplified models that are tractable to mathematical analysis. Both synaptic and intrinsic slow currents are incorporated in these models. 4. Rigorous and formal mathematical analysis of the above reduced models with particular attention to maintaining quantitative similarity to the full equations. He extends his model of vasomotion in collaboration with Dr. Jose Gonzalez-Fernandez to incorporate spatial aspects and coupling via the tissue bed. The goal is an explanation for the classic results of Krogh on recruitment of capillaries and to understand the dynamic regulation of blood flow at the microvascular level. He and Gonzalez-Fernandez intend to compare the results with some recent data of Segal's as well as older studies. As a side benefit, they hope to develop some new methods for analyzing systems of oscillators coupled though spatially continuous, but nonoscillatory media. He continues his collaboration with George Oster, using prior results on linear molecular motors and rotary motors to develop and analyze models of (i) the p ortal protein of bacteriophages, (ii) proton-driven motors, (iii) proton pumps. He collaborates with Leah Edelstein-Keshet on models for distribution of the lengths of actin filaments and their spatial variation within the cell. This work attempts to apply mathematical methods to aid in the understanding of physiological rhythms. Many human biological processes are governed by waxing and waning of different quantities. Some examples are the voltage of the heart pacemaker, the regular drifting in and out of different brain states during sleep and the control of oxygen flow in muscles by rhythmic contractions of small arteries during periods of exercise. The approach of this work is to use computer and mathematical models. Computing tools and tools for the visualization of simulations are developed as part of this work. The main goal is to use the models to bridge the gap between details about the microscopic workings of cells and their consequences for behavior. For example, when muscles work hard, they consume oxygen and thus need more oxygen from the arteries. Within the cells that form the arteries are special gates that sense the amount of "fuel" so that as this fuel decreases, the gates tell the artery to open up. This feedback loop provides a way to dynamically alter the rhythm of the artery in order to insure that the tissue is evenly oxygenated. Thus, by starting at the microscopic level of these gates or channels, a model is built that can explain how the consumption of oxygen is regulated in tissues. Another example occurs in the nervous system. Information from different parts of the brain is transmitted through special connections between nerve cells called synapses. Reaction times and cognitive abilities depend on the speed of this propagation as well as the prevention of propagation into other regions of the brain. Much is known about the reactions of individual nerve cells to stimuli. Models provide a way of coupling many such cells together in order to understand the behavioral consequences of the individual cell properties. Is the whole greater than the sum of the parts? The goal of this proposal is to try to answer this question for a variety of rhythmically controlled physiological processes.