Oscillations and Waves in Conductance-Based Neuronal Network Models

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Rubin0108857 This investigator develops and applies mathematicaltechniques for the study of activity patterns inconductance-based neuronal network models incorporating features,such as continuous coupling, diffusive interactions, andheterogeneity in parameter values, that have largely beenneglected in previous mathematical work on these systems. Manyof the mathematical results relate to the derivation andsubsequent analysis of subsystems of equations capturing dynamicson disparate time scales. One challenge in this endeavor is thatthe additional features considered prevent the application ofknown methods that could otherwise be used to reduce thesubsystem sizes and thereby facilitate analysis. The particularactivity patterns that are investigated are sustained, localizedactivity of cell populations in the thalamus involved in sleeprhythms and in tracking head direction; wave propagation, block,and reflection in inhomogeneous neuronal media such as branching,thickening, or damaged neurons; and periodic and chaoticoscillations in a heterogeneous population of respiratorypacemaker cells in the brain stem. While this work is motivatedby particular neuronal systems, it provides new steps indirections with broad relevance. Thus, the results of thisproject yield insight into pattern formation and selectionmechanisms that apply in specific neuronal contexts but that alsoconstitute more generally occurring phenomena. Neuronal networks of the central nervous system cantypically display multiple types of rhythmic activity, rangingfrom periodic bursts of synchronized oscillations to travelingwaves of activity to more disorganized patterns. Many of theseforms of activity appear to be relevant to fundamental neuronalprocesses. The overall goal of the investigator is to studymathematically the oscillatory population rhythms and wavepropagation arising in certain models of neuronal networks, basedon systems involved in such disparate tasks as navigation,respiration, and the generation of sleep rhythms. This workelucidates the mechanisms that allow these networks to support,and to modulate between, distinct activity patterns. Theseresults lead to conclusions about the roles of variousbiophysical parameters in shaping network dynamics, withindividual contributions teased apart in a way that may be quitedifficult to achieve with a purely experimental approach. Suchfindings can then serve as guideposts for future biological andcomputational experiments.