School of Science Department of Mathematics 52 Modeling Neural Circuit Dynamics in Whole-Brain Imaging Data of Larval Zebrafish Supervisor: HU Yu / MATH Student: CHAU Yu Hei / DSCT ZHANG Ke / MATH-PMA Course: UROP2100, Fall UROP2100, Fall The purpose of this report is to document the analysis we have conducted in the UROP project in the Fall semester (2020-2021). There have been two main areas of focus: the zebrafish prey capture analysis part and the gating model part. Our methods are by no means novel. Most of the analysis are conducted using existing methods, for example, by using available software packages such as MATLAB. Some of them are technical in nature, such as visualizing neuronal information using a 2D image plot. There are also reproduction of results established in other papers, so as to test hypotheses with the given models. The relation between some of the analyses below may be small, as they are minor fragments inside the zebrafish project. This report consists of sections that documents a particular aspect in the project. In every section of this report, the individual goals of the corresponding section will first be stated, and then the details such as methodology and usage will be written. RNN for Biological Neural Networks Supervisor: HU Yu / MATH Student: CHEN Zixin / DSCT Course: UROP1100, Fall Artificial Recurrent Neural Networks such as LSTM and GRU are highly versatile and achieve breakthrough performance in many applications of sequential data. However, the RNN structure for neural science is different from the common one in other fields. Thus, in this project, we customize a RNN for neural science and use it to model the complex temporal dynamics of real neural networks in the brain and explore variations of RNN that are biological plausible with interpretable parameters that can be potentially linked to anatomical structures and biophysical properties. After integrated three Bio-RNN related papers, we built a special RNN and explored its properties on three tasks related to neural science. The results of these experiments prove the efficiency of our RNN. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: CHOY Sin Hang Sonia / MATH-IRE Course: UROP2100, Fall Cluster algebras have been linked to snake graphs, continued fractions and surface triangulations in separate papers. At the heart of cluster algebra is the mutation of a cluster variable, where one can mutate one variable to obtain a new one. This report establishes the correspondences between cluster algebras and positive continued fractions, and the effect of a cluster mutation on the corresponding continued fraction, via their links with triangulations and snake graphs with weighted edges.