CAREER: Random Walk in Random Environment

University Of Utah, Salt Lake City UT

Investigators

Abstract

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 the understanding 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.