One of the important challenges in theoretical neurobiology is understanding the relationship between spatially structured activity states in the brain and the underlying neural circuitry that supports them. This has led to considerable interest in analyzing reduced biological models of neuronal networks. Most analytical studies of these network models assume that the system is spatially homogeneous. Recently, however, the principal investigator has shown that the combined effect of a spatially localized inhomogeneous input and recurrent synaptic interactions between neurons can result in nontrivial forms of coherent oscillations and waves. This motivates the current research project, which will carry out a more detailed study of the cellular and network mechanisms underlying the generation of these oscillations and waves. The research program will be divided into three parts corresponding to three distinct neurobiological application areas: (I) epileptiform activity in a model of disinhibited neural tissue, (II) stimulus-induced coherent oscillations in a model of primary visual cortex, and (III) localized activity states in a two-layer thalamic network model of the head direction system. In each of these cases the existence and stability of coherent activity states will be analyzed, and their dependence on various biologically relevant parameters will be determined. The mathematical aspects of the work will also be applicable to other population-based biological systems, in which the basic elements at the molecular, cellular or organismal level interact nonlocally in space.
Analysis of the dynamical mechanisms underlying spatially structured activity states in neural tissue is crucially important for understanding a wide range of neurobiological phenomena, both naturally occurring and pathological. For example, neurological disorders such as epilepsy and migraine are characterized by waves propagating across the surface of the brain. Determining the various cellular and network properties underlying the onset of such disorders could ultimately help in developing clinical techniques for eliminating them. Spatially coherent activity states are also prevalent during the normal healthy functioning of the brain, encoding local properties of visual and auditory stimuli, encoding head direction and spatial location, and maintaining persistent activity states in short-term working memory.