Mathematical analysis of dynamic activity patterns in neuronal network models

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

Rubin The investigator develops and applies mathematical techniques for the study of activity patterns in conductance-based neuronal network models, which include biophysically relevant time scales and parameters, and in activity-based models, in which the temporal variation in the cumulative activity of a neuronal population is tracked with a single equation. The models analyzed include features, such as spatially continuous coupling, temporally varying input patterns, and heterogeneity in parameter values, that have largely been neglected in previous mathematical work yet which are relevant in a wide range of neuronal systems. The mathematical methods developed relate to the derivation and subsequent analysis of subsystems of equations capturing dynamics on disparate time scales. Further, the study of biophysically-based models is complemented by the development of a new and broadly applicable inverse approach, in which observations of network dynamics is used to constrain possible patterns of connections between cells in the network in cases where this connectivity has not been experimentally determined. The mathematical study of neuronal network models in this project is motivated by experimental data on changes in neuronal rhythms between normal and parkinsonian conditions; by activity of neuronal populations involved in navigation; and by the behavior of neurons that help to drive respiration. In each of these settings, networks of neurons perform particular tasks by engaging in, and switching between, organized firing patterns. Correspondingly, this project elucidates the mechanisms that allow the relevant networks to support, and to modulate between, particular modes of activity. These analytical results lead to general conclusions about how various components combine to shape neuronal network dynamics, with their particular contributions teased apart in a way that would be difficult to achieve with a purely experimental approach. The understanding gained is well suited to guide future biological and computational experiments. In particular, this project promotes an interdisciplinary partnership with J. Smith's lab at NIH, which allows for direct access to experimental data and testing of model predictions. Students engaged in the work receive interdisciplinary training in an important area of interest to both mathematics and biology.