Complex and Chaotic Biological Dynamics

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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.****