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

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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.