The first major goal of this work is to learn how certain brain regions (olfactory system, hippocampus, and cerebellum) learn very complex stimuli that employ a combinatorial code to identify stimuli as points in a high-dimensional space. For example, the simple fruit fly olfactory system uses the firing rates of 50 different types of odorant receptors to identify each odor by placing it at a point in a SO-dimensional space. Although the fly olfactory system is well understood, less is known about analogous regions in vertebrate brains, and our goal is to begin to learn about these other regions. The first step is to start with the mouse olfactory system that is similar to the fly in some ways but has complexities that are absent in insects. These complexities include an enhanced ability to handle noise in the odor and to learn over time to discriminate between very similar odors (e.g., two types of red wines). Preliminary evidence shows that it should be possible to learn the role of these complexities in vertebrate olfaction. The research design involves studying the anatomy and recording the firing rates of different types of neurons at different levels of the mouse olfactory system and in applying computational methods and algorithms that have proved successful in earlier work to describe these complexities. The second major goal is to use insights into how these brain regions operate to improve the function of computer algorithms. For a long time, a dream of many neuroscientists and computer scientists has been to understand how the brain works well enough that we could translate insights from the brain to improve machine computation. Indeed, experience has shown that the brain has evolved novel variations of information processing algorithms used by computer scientists to solve general computational problems. With sufficient insight into algorithms used by the brain, these insights may provide unexpected ways to improve the function computer science algorithms. Further, understanding the circuit mechanisms involved in olfactory processing can help illuminate the basis of a variety of smell disorders, and may in the future lead to the construction of artificial smelling devices.