| This research has developed a robust framework for understanding, predicting, and controlling the collective emergent behavior of complex rhythmic systems. This framework was based on a relatively new type of mathematical approach for modeling rhythmic systems known as phase modeling. The key feature of this approach was that it works directly with the phase of an oscillation, rather than the underlying physiochemical phenomena which causes the oscillation, dramatically simplifying the mathematical analysis of rhythmic systems.;Experimental methods were developed to measure the dynamical properties of rhythmic elements from their observable macroscopic behaviors with high accuracy. These experimental measurements were used to construct phase models which were subsequently used to described the collective behavior of a large population of rhythmic elements. Experiments conducted over the course of this research on oscillatory electrochemical systems indicate that phase models provide a quantitatively accurate description of the collective behavior of rhythmic systems. Experiment-based phase models were utilized to determine a global, nonlinear, time-delayed feedback stimulation which would produced a desired collective behavior within a target rhythmic system. Experimental results illustrate that collective behavior such as phase synchronization, phase clustering, itinerate clustering, and desynchronization can be engineered into a rhythmic electrochemical system by manipulating the parameters which control the feedback interactions between elements.;Experiments have been conducted to apply phase models to complex rhythmic systems such as phase coherent chaotic systems, diffusively coupled systems, groups of rhythmic elements, and biological neurons. Applications of this work may be found in the field of deep brain stimulation for the treatment of neurological diseases such as epilepsy. |
