CAREER: Random Walk in Random Environment

PI将继续他的工作，开发技术，以解决随机介质中随机运动领域的重要问题，如不变性原则，大偏差等。此外，PI打算通过将扩散过程理论应用于计算神经科学来扩大他的研究的影响。作为第一步，PI建议研究广泛使用的带噪声和自适应的集成-发射模型。主要问题包括响应恒定或周期性变化输入的平稳解的存在性和稳定性。这项建议也有教育方面的内容；见下文。传统上，许多物理系统都是用扩散过程建模的。通常情况下，发生扩散的介质具有一定的不均匀性（例如，电子在介质中运动，波在地下岩石中扩散，等等）。这一特性的包含使模型更加真实，并揭示了在恒定或周期性介质中不存在的各种新效果。PI工作的主要目标是进一步了解这种疾病在媒介中的后果。PI的另一部分研究源于他的信念，即在生物研究的爆炸式增长中，数学科学需要更强的参与。特别是，神经科学中使用的神经网络模型的复杂性使得它们的建模在计算和分析上具有挑战性。最近有一种分析这类网络的技术，可以把种群动态看作是一个特定的扩散过程。PI建议将这种技术扩展到更广泛的神经元模型，包括反馈（适应）机制。研究结果将应用于特定的神经生物学问题，如运动声源的定位和视觉中的对比度适应。提案的第三个组成部分是教育。PI将开发新课程，举办研讨会和研讨会，让当地崭露头角的科学家参与，组织本科生的研究项目，并积极让研究生和博士后参与他正在进行的研究。

The PI will continue his work on developing techniques to address important problems in the field of random motion in random media, such as invariance principles, large deviations, etc. In addition, the PI intends to broaden the impact of his research by also applying the theory of diffusion processes to computational neuroscience. As a first step, the PI proposes to study the widely-used integrate-and-fire model with noise and adaptation. The primary questions include existence and stability of stationary solutions in response to a constant or periodically varying input. The proposal also has an educational component; see below.Traditionally, many physical systems are modeled with a diffusion process. It is often the case that the medium in which the diffusion occurs possesses certain inhomogeneities (e.g. an electron moving in an allow, waves diffusing in rocks underground, etc). The inclusion of this property makes the model far more realistic and reveals a wide variety of new effects, not present in constant or periodic media. The main goal of the PI's work is to further theunderstanding of the consequences of this disorder in the medium. Another part of the PI's research stems from his belief in the need for a stronger involvement of mathematical sciences in the explosive growth of biological research. In particular, the complexity of neuronal network models used in Neuroscience makes their modeling a computationally and analytically challenging task. A recent technique for analyzing a class of such networks allows to view the population dynamics as a certain diffusion process. The PI proposes to extend this technique to a wider class of neuronal models, including a feedback (adaptation) mechanism. The results will be applied to specific neurobiological problems, such as localization of moving sound sources and contrast adaptation in vision. The third component in the proposal is educational. The PI will develop new courses, run seminars and workshops involving local budding scientists, organize undergraduate work on research projects, and actively involve graduate students and postdoctoral fellows in his ongoing research.