Mathematical Sciences: Applications of Mathematics in Neurobiology

研究者和他的同事们使用数学和计算工具来研究耳蜗背核（DCN）和脑干下部外侧上橄榄（LSO）对听觉信息的处理。研究DCN的目的是了解DCN如何从复杂的声音中提取频谱信息。研究LSO的目的是了解它通过使用耳间强度（和时间）差异来计算和编码方位声定位的机制。这两个项目都是基于现实的神经生理学和神经解剖学，目的是了解大群细胞是如何共同进行计算的。此外，对营养发育方案相对于正向随机方案的不稳定性的明显稳定性背后的原因进行了数学调查。研究者还研究了反应-双曲型偏微分方程的性质。这种类型的方程出现在神经轴突（和其他真核细胞）的物质运输中。以前的建模研究已经提出了关于反应-双曲型偏微分方程系统中近似行波的存在性和行为的数学问题。利用奇异摄动技术，将前人关于在化学速率常数变大的极限下存在近似行波的特殊线性情况的结果推广到非线性情况。利用概率方法导出了该渐近极限下的闭形式近似解公式。由于人们对概率技术如何应用于线性和非线性双曲系统（与椭圆或抛物系统相反）知之甚少，这项工作可能对其他双曲系统有重要的应用。研究人员使用数学和计算工具来研究脑干细胞对审计信息的处理。该项目的目的是了解大群神经细胞如何协同工作来进行计算，例如在空间中定位声音或通过分析声音中包含的频率模式来识别声源。这不仅让我们深入了解我们的听觉和大脑是如何工作的，而且还应该为在机器人控制系统中执行类似功能的技术设备的设计提供建议。研究人员还研究了一类在神经轴突的物质运输中出现的偏微分方程。这些方程式不仅使人们能够更好地理解物质是如何在正常的神经细胞中运输的，而且还能深入了解在某些工作场所毒素存在和某些疾病（如阿尔茨海默病）中发生的运输中断。

Reed The investigator and his colleagues use mathematical and computational tools to investigate processing of auditory information in the dorsal cochlear nucleus (DCN) and the lateral superior olive (LSO) of the lower brainstem. The purpose of the study of the DCN is to understand how the DCN functions to extract spectral information from complex sounds. The purpose of the study of the LSO is to understand the mechanisms by which it computes and encodes azimuthal sound location by using interaural intensity (and time) differences. The purpose of both projects, which is based on realistic neurophysiology and neuroanatomy, is to understand how large groups of cells collectively perform calculations. In addition, a mathematical investigation of the reasons underlying the apparent stability of trophic developmental schemes as opposed to the instability of forward stochastic schemes is carried out. The investigators also investigate the properties of reaction-hyperbolic partial differential equations. Equations of this type occur in the transport of materials in nerve axons (and other eukaryotic cells). Previous modelling studies have raised mathematical questions about the existence and behavior of approximate travelling waves in reaction-hyperbolic systems of partial differential equations. Singular perturbation techniques are used to extend to the nonlinear case previous results on special linear cases where approximate travelling waves have been shown to exist in the limit where the chemical rate constants get large. Probabilistic methods are used to derive closed form approximate solution formulas in this asymptotic limit. Since little is known about how probabilistic techniques can be used in linear and nonlinear hyperbolic systems (as opposed to elliptic or parabolic systems), this work may have important applications to other hyperbolic systems. The investigators use mathematical and computational tools to investigate the processing of audit ory information by cells in the brainstem. The purpose of the project is to understand how large groups of nerve cells work together to perform calculations such as localizing sounds in space or recognizing the sound source by analyzing the pattern of frequencies contained in the sound. This not only gives insight into how we hear and how the brain works, but should also suggest designs for technological devices that perform similar functions in the control systems of robots. The investigators also investigate a class of partial differential equations that arise in the transport of materials in nerve axons. These equations not only allow one to better understand how materials are transported in normal nerve cells, but also give insight into the breakdown of transport that occurs in the presence of some workplace toxins and in certain diseases such as Alzheimer's disease.