Complex and Chaotic Biological Dynamics

生物和生理系统表现出丰富的复杂动力学。以前的工作演示了**相对简单的理论模型如何解释控制良好的模型实验**系统中的复杂动力学。尽管取得了这些成功，但对大多数自然发生的生物节律的深入理解还没有**实现。尤其是在理解基因、心脏和神经**网络的动力学方面出现了一些基本问题。在我的研究中，我为大量相互作用的元素建立了理论模型，并研究了**动力学如何依赖于可修改的实验参数的变化，包括空间**异质性、噪声、时间延迟、网络结构。*近年来见证了新的实验数据收集方式的发展，使我们**能够收集大量数据。这些新方法包括使用荧光染料对活组织进行光学成像、基因表达芯片、多电极记录、DNA测序。作为这些进步的结果，人们越来越认识到对生物系统进行数学分析和建模的必要性。目前的工作将集中于涉及不同领域中复杂动力学和几何结构之间的联系的问题，包括：包含起搏器和可兴奋细胞的心脏组织培养中的节律产生；皮层网络的视觉信息处理；无脊椎动物发育过程中的基因控制。这些领域由于存在基本的未解决的科学问题而团结在一起，在我看来，这些问题将需要理论和实验相结合的跨学科方法。此外，所有这些系统在空间中都包含复杂的网络，其中的基本数学性质仍然需要阐明。通过提供对正常动力学的新理解，这项工作可能为控制异常动力学的方法提供新的见解。参与这些项目的学生将接受生物和数学方面的跨学科培训，这将使他们处于新兴科学领域的前沿。

Biological and physiological systems display a wealth of complex dynamics. Previous work demonstrated how**comparatively simple theoretical models account for complex dynamics in well-controlled model experimental**systems. Despite these successes, a deep understanding of most naturally occurring biological rhythms has not**been achieved. In particular fundamental issues arise in understanding dynamics in gene, cardiac, and neural**networks. In my research, I develop theoretical models for large numbers of interacting elements, and study**how the dynamics depends on variations on modifiable experimental parameters including spatial**heterogeneity, noise, time delays, network structure.****Recent years have witnessed the development of new modalities of experimental data collection that enable us ** to gather large amounts of data. These new methods include optical imaging of living tissue using fluorescent dyes, gene expression chips, multielectrode recordings, DNA sequencing. As a consequence of these advances, there is growing recognition of the necessity of mathematical analysis and modeling of biological systems. The current work will focus on problems that involve connections between complex dynamics and geometric structures in diverse areas including: rhythm generation in cardiac tissue culture containing both pacemakers and excitable cells; visual information processing by cortical networks; gene control during development in invertebrates. These areas are united by the presence of fundamental unsolved scientific questions that will, in my opinion, require interdisciplinary approaches combining theory and experiment. Further, all these systems contain complex networks in space in which basic mathematical properties still need to be elucidated. By providing new understanding of normal dynamics, the work may provide new insights into methods to control abnormal dynamics. Students engaged in these projects will receive interdisciplinary training in both biology and mathematics that will place them at the forefront of emerging fields of science.****