Methods from nonlinear dynamics are used to study the coupling between oscillators and non-oscillators in model neuronal networks. Specifically, averaging is used to analyze the termination of waves in spatially connected networks by reducing the complicated conductance-based models to scalar spatial models. Phase-plane methods are applied to the reduced systems. Averaging is also used at the single spike level to understand the transition between synchrony and asynchrony in coupled networks that have different types of connections. Population density methods and linearized stability analysis will distinguish between the onset of synchrony, clusters, and propagating waves in spatially distributed networks of neural oscillators. Indirect coupling between excitable and oscillatory cells will be analyzed using a combination of phase-resetting curves and the dispersive properties of excitable cables.
When neurons are connected together, they can often produce persistent activity. Such persistent activity has been implicated in short-term memory -- how an animal or human "holds a thought." What kinds of interactions disrupt this and which are necessary to maintain the activity are some of the questions that are asked in this proposal. When activity is too persistent then certain pathologies arise such as epilepsy. Thus, one goal of this proposal is to understand how to strike a balance between the ability to produce stable persistent activity while preventing its pathological propagation into quiet regions. When neurons fire they communicate with other neurons and depending on the interactions, the result can be that the neurons want to fire together or they want to fire asynchronously. The latter is useful for persistent activity. Synchrony on the other hand is crucial for several normal physiological processes. For example, it is known that certain cells in the base of the brain organize the output of hormones. The electrical activity of these cells is synchronized yet the mechanisms for this synchrony remain unknown since there are no direct connections between the individual oscillating neurons. We will study mechanism through which indirect coupling can produce synchronous behavior. Tools and methods developed in this proposal will have applications well beyond neuroscience, as the questions of synchrony and propagation of "information" are ubiquitous in biology from the single cell level on up to the ecological interactions between populations.
