Complex and Chaotic Biological Dynamics

生物和生理系统显示出丰富的复杂动力学。先前的工作证明了在控制良好的模型实验系统中，相对简单的理论模型如何解释复杂的动力学。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刻理解尚未实现。特别是在理解基因、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，我开发了大量相互作用元素的理论模型，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。****近年来见证了实验数据收集新模式的发展，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。目前的工作将集中在涉及复杂动力学和几何结构之间的联系的问题，在不同的领域，包括：心律生成的心脏组织培养包含起搏器和兴奋细胞；皮层网络对视觉信息的处理；无脊椎动物发育过程中的基因控制。这些领域是由尚未解决的基本科学问题联系在一起的，在我看来，这些问题需要跨学科的方法来结合理论和实验。此外，所有这些系统都包含空间中的复杂网络，其中的基本数学性质仍然需要阐明。通过对正常动力学的新认识，可以为异常动力学的控制方法提供新的见解。参与这些项目的学生将接受生物学和数学方面的跨学科培训，这将使他们站在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.****