Guckenheimer 9705780 The investigator studies dynamical systems with multiple time scales as models of neural systems. Data and models from a small invertebrate neural network, the stomatogastric ganglion of lobsters, guide the work and provide case studies. Phenomena observed in the stomatogastric ganglion, such as bursting oscillations and spike frequency adaptation, can best be modeled as dynamical systems with multiple time scales. Previous mathematical analyses of qualitative properties of multiple time scale dynamical systems have dealt mainly with local phenomena that occur in low dimensions. The dynamics of neural systems raise questions that are not addressed by existing theories of multiple time scale systems. The aim here is to extend the theory by classifying qualitative features of the global dynamics and bifurcations for systems with two time scales. This is an ambitious endeavor to extend theories of nonlinear dynamical systems and singularly perturbed systems of ordinary differential equations that have been developed over the past thirty years. Using numerical investigations as a guide, the dictionary of patterns that occur in this setting is described and their analytical properties are characterized. This work draws heavily upon the theories of bifurcations of dynamical systems and models of hybrid dynamical systems that combine continuous and discrete components. As needed, numerical algorithms are developed that facilitate the simulation and analysis of multiple time scale systems. The initial emphasis of the mathematical work is upon systems that have two slow variables and two fast variables. Numerical investigations of conductance-based models for the stomatogastric ganglion also are performed. The results of these studies are compared with data and used to guide the refinement of the models. Nervous systems of animals regulate and control muscular activi ty such as locomotion. Well developed theories enable the construction of models for the electrical properties of nerve membranes in these processes, but there is little understanding of the dynamical principles used by organisms. This project investigates dynamical models of a small neural system consisting of fourteen neurons that control rhythmic motions of the foregut of lobsters. This system is used because it is small enough that unique properties of each neuron within the system have been identified, but large enough that the network architecture of the interactions among neurons is also important. The system displays a rich repertoire of rhythmic behavior. The focus of this project is on features of the behavior that involve different time scales. Mathematical theories have been successful at describing universal properties of the dynamics observed in an astounding array of physical and natural systems, but these theories need to be extended to systems with more than one time scale. The goal is to construct classifications of dynamical patterns that are the components from which the complex behaviors of neural systems are formed. Models of these systems are also complex. Computational investigations are required to predict their dynamical behavior. This project seeks to implement algorithms that improve our ability to extract useful information from the models and guide the improvement of their fidelity. Both the theoretical analysis and the numerical methods that are developed encompass all models of dynamical systems with multiple time scales and can be used far beyond the context of the neural system that is the focus of this study.
