Mathematical Sciences: Mathematical Methods for Biological Rhythms and Population Genetics

研究人员研究了两个不同但协同作用的领域 应用数学：神经网络和群体遗传学。 神经网络的研究是基于电压控制 振荡器模型神经元（VCON），可构建 具有有趣的数学模型的电子设备， 与生理神经细胞有许多相似之处。 调查人员 最近推出了VCON网络，过滤，处理， 基于频率编码存储信息;它们扩展并 与已知的生理回路并行地完善这项工作。 在这方面感兴趣的数学方面是旋转向量 和动力系统的广义傅立叶分析。 的 应用方面与电路的设计和研究有关 可以复制哺乳动物大脑部分的已知功能 基于频率编码和解码。 具体而言是 研究人员研究了音调拓扑映射，存储和检索 消息，以及处理电路的结构和功能 类似于新皮质柱。 本建议的第二部分 研究人口和遗传学。 调查人员专注于 食物链和细菌群落中的菌落同步， 细菌菌苔中的模式形成，以及 基因工程微生物 计算方法 的VCON网络的模拟，研究 细菌群落 该项目涉及数学生物学中的两个问题 这说明了数学的两个重要贡献 对生命科学的贡献：首先，数学使我们能够 分析类似于我们大脑部分的大型网络， 第二，数学公式使我们能够使用科学的 计算机来预测实验的结果， 昂贵或难以执行。 首先，调查人员 研究大量模仿大脑的电子设备 电路. 这些器件（称为VCO）是基于锁相 在电子通信中广泛使用的环路。 他们 我设计、分析和模拟了VCON网络， 注意力集中在许多竞争刺激中最持久的， 记忆和回忆信息。 他们将这项工作扩展到 包括大脑的其他方面，包括大脑的某些部分， 海马体。 这项工作的新颖之处在于它是基于 频率编码信息而不是离散信息 位，如数字计算机中的位。 因此，网络 相当稳定，即使部分损坏也能正常工作。 在 这项工作第二部分，他们使用为VCON开发的方法 研究细菌群落问题的网络。 而 这两个问题领域相距甚远， 两者的数学方面非常相似：细菌 导致振荡的生活，并使用大型网络的方法 的VCON振荡器，研究人员能够描述在 用数学术语来说， 在土壤中生长。 除了解决那些 这两个领域之间的相似之处，他们研究的具体问题， 细菌遗传学，包括基因稳定性 细菌细胞系的工程变异。

The investigators study two different but synergistic areas of applied mathematics: neural networks and population genetics. The neural network studies are based on Voltage Controlled Oscillator model Neurons (VCONs), which are constructible electronic devices that have interesting mathematical models and many parallels with physiological nerve cells. The investigators have recently derived VCON networks that filter, process, and store information based on frequency encoding; they extend and refine this work in parallel with known physiological circuits. The mathematical aspects of interest in this are rotation vectors and generalized Fourier analysis of dynamical systems. The applied aspects are related to the design and study of circuits that can reproduce known functions of parts of mammalian brains based on frequency encoding and decoding. Specifically, the investigators study tonotopic mappings, storage and retrieval of messages, and the structure and function of processing circuits similar to neocortical columns. The second part of this proposal studies populations and genetics. The investigators focus on food chains and colony synchronization in bacterial communities, pattern formation in bacterial lawns, and plasmid stability in genetically engineered microorganisms. Computational methods developed for simulation of VCON networks are applied to study bacterial communities. The project involves two problems in mathematical biology that illustrate two important contributions that mathematics makes to the life sciences: First, mathematics enables us to analyze large networks that resemble parts of our brains, and second, mathematical formulations enable us to use scientific computers to predict the outcome of experiments that are expensive or difficult to perform. First, the investigators study large collections of electronic devices that mimic brain circuits. These devices (called VCONs) are based on phase-locked loops that are widely used in electronic communications. They have designed, analyzed, and simulated VCON networks that focus attention on the most persistent among many competing stimuli, and that memorize and recall messages. They extend this work to include other aspects of the brain, including parts of the hippocampus. The novelty of this work is that it is based on frequency encoded information rather than discrete information bits, as in digital computers. As a result, the networks are quite stable, even functioning well when partially destroyed. In the second part of this work, they use methods developed for VCON networks to study problems about bacterial communities. While these two problem areas are widely separated, certain mathematical aspects of the two are remarkably similar: Bacteria lead oscillatory lives, and using the methods for large networks of VCON oscillators, the investigators are able to describe in mathematical terms large communities of bacteria -- for example, as arising in soil. In addition to working on problems that are similar between the two areas, they study problems specific to bacterial genetics, including how stable are genetically engineered variations in bacterial cell lines.