Differential and Integral Equations in Neurobiology

点击翻译按钮获取中文摘要

The investigator studies the dynamics of certain classes ofintegro-differential equations that arise in models of cortex. Heexamines how recurrent activity terminates due to any of severaleffects such as synaptic depression, depolarization block, andafter-hyperpolarization. He also considers how recurrentconnections lead to bistability and under what conditions this issufficient to produce localized spatial states, wave fronts, andpulses. He uses methods of averaging and bifurcation theory toderive and analyse simplified dynamics from these models. Themathematical results explain a variety of experimental findingsin both normal and pharmacologically manipulated neural tissue.These methods are applied to the Limax procerebral lobe, theturtle olfactory bulb, and various cortical slice preparations. Neurons, the main cells in the brain, are responsible forprocessing inputs from the outside world and converting this tomotor actions. They communicate by producing spikes that arepropagated to different regions of the brain by chemicals calledtransmitters. The nvestigator is interested in how thiselectrical and chemical information moves from one part of thebrain to others. He uses simulations and mathematics to deriveequations and general principles about how this is done.Experiments have shown that during some cognitive tasks, certaingroups of neurons fire synchronously. He is interested in theconditions that are necessary to attain precise timing betweendifferent groups of neurons. In order to study these, he hasdeveloped mathematical software for the study of complexbiological and physical systems. As part of this project, hecontinues to develop this software and to make it widelyavailable. The project also provides interdisciplinary trainingopportunities for students and postdocs.