Mathematical analysis of dynamic activity patterns in neuronal network models

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Rubin The investigator develops and applies mathematical techniques forthe study of activity patterns in conductance-based neuronal networkmodels, which include biophysically relevant time scales andparameters, and in activity-based models, in which the temporalvariation in the cumulative activity of a neuronal population istracked with a single equation. The models analyzed include features,such as spatially continuous coupling, temporally varying inputpatterns, and heterogeneity in parameter values, that have largelybeen neglected in previous mathematical work yet which are relevant ina wide range of neuronal systems. The mathematical methods developedrelate to the derivation and subsequent analysis of subsystems ofequations capturing dynamics on disparate time scales. Further, thestudy of biophysically-based models is complemented by the developmentof a new and broadly applicable inverse approach, in whichobservations of network dynamics is used to constrain possiblepatterns of connections between cells in the network in cases wherethis connectivity has not been experimentally determined. The mathematical study of neuronal network models in this projectis motivated by experimental data on changes in neuronal rhythmsbetween normal and parkinsonian conditions; by activity of neuronalpopulations involved in navigation; and by the behavior of neuronsthat help to drive respiration. In each of these settings, networksof neurons perform particular tasks by engaging in, and switchingbetween, organized firing patterns. Correspondingly, this projectelucidates the mechanisms that allow the relevant networks to support,and to modulate between, particular modes of activity. Theseanalytical results lead to general conclusions about how variouscomponents combine to shape neuronal network dynamics, with theirparticular contributions teased apart in a way that would be difficultto achieve with a purely experimental approach. The understandinggained is well suited to guide future biological and computationalexperiments. In particular, this project promotes aninterdisciplinary partnership with J. Smith's lab at NIH, which allowsfor direct access to experimental data and testing of modelpredictions. Students engaged in the work receive interdisciplinarytraining in an important area of interest to both mathematics andbiology.