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What distinguishes living neural systems from many man-maid devices for information processing is their remarkable ability to be highly adaptable to environment and amenable for control and information processing and, at the same time, to be reasonably stable. The loss of this ability would unavoidably lead to disease. In this talk I will discuss my present research in the area of mathematical biology of mammalian basal ganglia - brain nuclei, which (among other things) control motor programs and are impacted in Parkinson's disease. Basal ganglia networks constitute a nice example of many neural systems, which are intimately involved in the normal human behavior; nevertheless their basic principles of operation are poorly understood. I present my developments of the mathematical models of basal ganglia circuitry, which allow for mathematical and computational exploration of brain function, and the results of the studies of synchrony in the brain, which may serve as basis for the application of dynamical systems theory to the analysis of brain. I will discuss the implications obtained from modeling and real-data analysis and their relevance to both general problems of neural networks dynamics (mechanisms of basal ganglia, which may be generic for living neural networks)and to specific clinical issues (pathophysiology of Parkinson's disease). I will pay attention to the general problems of basal ganglia dynamics and dynamics of other living neural networks, which still wait to be answered. I will talk about possible approaches to the search for the answers and possible answers we may expect to obtain.