The investigator studies the dynamical behavior of systems of Wilson-Cowan type integro-differential equations that arise in modelling cortex. His goal is to understand the subtle connections between neurons in two-dimensional neural sheets, which are responsible for (i) the formation of stable rotating waves, and (ii) the existence of organizing centers that cause periodic ring-shaped waves to form and propagate outward through the medium. He employs computational techniques, as well as asymptotic analysis and reduction methods, which allow the models to simplfy to tractable forms. The mathematical and computational results predict and verify the existence of stable rotating spirals, and periodic ring-shaped waves recently discovered in tangential slices of occipital cortex of rat brain. The techniques developed in this study provide a new direction for understanding persistent pattern formation in general neural network models. Neuronal networks in the central nervous system exhibit a variety of persistent wave patterns. These range from oscillatory waves consisting of synchronized bursts of spikes to two-dimensional ring-shaped waves and rotating spirals. The investigator is interested in understanding how such two-dimensional waves form and propagate in the brain. He uses simulations and mathematical methods to develop models that aid in the understanding of wave formation. Cooperative research between the investigator and experimental neuroscientists has led to the recent discovery of spiral waves in rat cortex. This is the first true verification of spiral waves in mamallian cortex. Spiral waves are conjectured to be associated with the occurence of seizures in the brain. The results of this project are also included in the design of new interdisciplinary courses for students.
