Oscillations and Waves in Conductance-Based Neuronal Network Models

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

Rubin 0108857 This investigator develops and applies mathematical techniques for the study of activity patterns in conductance-based neuronal network models incorporating features, such as continuous coupling, diffusive interactions, and heterogeneity in parameter values, that have largely been neglected in previous mathematical work on these systems. Many of the mathematical results relate to the derivation and subsequent analysis of subsystems of equations capturing dynamics on disparate time scales. One challenge in this endeavor is that the additional features considered prevent the application of known methods that could otherwise be used to reduce the subsystem sizes and thereby facilitate analysis. The particular activity patterns that are investigated are sustained, localized activity of cell populations in the thalamus involved in sleep rhythms and in tracking head direction; wave propagation, block, and reflection in inhomogeneous neuronal media such as branching, thickening, or damaged neurons; and periodic and chaotic oscillations in a heterogeneous population of respiratory pacemaker cells in the brain stem. While this work is motivated by particular neuronal systems, it provides new steps in directions with broad relevance. Thus, the results of this project yield insight into pattern formation and selection mechanisms that apply in specific neuronal contexts but that also constitute more generally occurring phenomena. Neuronal networks of the central nervous system can typically display multiple types of rhythmic activity, ranging from periodic bursts of synchronized oscillations to traveling waves of activity to more disorganized patterns. Many of these forms of activity appear to be relevant to fundamental neuronal processes. The overall goal of the investigator is to study mathematically the oscillatory population rhythms and wave propagation arising in certain models of neuronal networks, based on systems involved in such disparate tasks as navigation, respiration, and the generation of sleep rhythms. This work elucidates the mechanisms that allow these networks to support, and to modulate between, distinct activity patterns. These results lead to conclusions about the roles of various biophysical parameters in shaping network dynamics, with individual contributions teased apart in a way that may be quite difficult to achieve with a purely experimental approach. Such findings can then serve as guideposts for future biological and computational experiments.