Mathematical Sciences: Applications of Mathematics in Neurobiology

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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.