Two-dimensional Models of Neural Sheets

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Troy The investigator studies the dynamical behavior of systemsof Wilson-Cowan type integro-differential equations that arise inmodelling cortex. His goal is to understand the subtleconnections between neurons in two-dimensional neural sheets,which are responsible for (i) the formation of stable rotatingwaves, and (ii) the existence of organizing centers that causeperiodic ring-shaped waves to form and propagate outward throughthe medium. He employs computational techniques, as well asasymptotic analysis and reduction methods, which allow the modelsto simplfy to tractable forms. The mathematical and computationalresults predict and verify the existence of stable rotatingspirals, and periodic ring-shaped waves recently discovered intangential slices of occipital cortex of rat brain. Thetechniques developed in this study provide a new direction forunderstanding persistent pattern formation in general neuralnetwork models. Neuronal networks in the central nervous system exhibit avariety of persistent wave patterns. These range from oscillatorywaves consisting of synchronized bursts of spikes totwo-dimensional ring-shaped waves and rotating spirals. Theinvestigator is interested in understanding how suchtwo-dimensional waves form and propagate in the brain. He usessimulations and mathematical methods to develop models that aidin the understanding of wave formation. Cooperative researchbetween the investigator and experimental neuroscientists has ledto the recent discovery of spiral waves in rat cortex. This isthe first true verification of spiral waves in mamallian cortex.Spiral waves are conjectured to be associated with the occurenceof seizures in the brain. The results of this project are alsoincluded in the design of new interdisciplinary courses forstudents.