Differential and Integral Equations in Neurobiology

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

The investigator studies the dynamics of certain classes of integro-differential equations that arise in models of cortex. He examines how recurrent activity terminates due to any of several effects such as synaptic depression, depolarization block, and after-hyperpolarization. He also considers how recurrent connections lead to bistability and under what conditions this is sufficient to produce localized spatial states, wave fronts, and pulses. He uses methods of averaging and bifurcation theory to derive and analyse simplified dynamics from these models. The mathematical results explain a variety of experimental findings in both normal and pharmacologically manipulated neural tissue. These methods are applied to the Limax procerebral lobe, the turtle olfactory bulb, and various cortical slice preparations. Neurons, the main cells in the brain, are responsible for processing inputs from the outside world and converting this to motor actions. They communicate by producing spikes that are propagated to different regions of the brain by chemicals called transmitters. The nvestigator is interested in how this electrical and chemical information moves from one part of the brain to others. He uses simulations and mathematics to derive equations and general principles about how this is done. Experiments have shown that during some cognitive tasks, certain groups of neurons fire synchronously. He is interested in the conditions that are necessary to attain precise timing between different groups of neurons. In order to study these, he has developed mathematical software for the study of complex biological and physical systems. As part of this project, he continues to develop this software and to make it widely available. The project also provides interdisciplinary training opportunities for students and postdocs.