Gamma and Theta Rhythms in Biophysical Models of Hippocampal Circuits

The neural circuits of the hippocampus are extremely complex, with many classes of interneurons whose contributions to network dynamics and function are still unknown. Nevertheless, reduced models can provide insight into aspects of the dynamics and associated function. In this chapter, we discuss models at a variety of levels of complexity, all simple enough to probe the reasons for the behavior of the model. The chapter focuses on the main rhythms displayed by the hippocampus, the gamma (30–90Hz) and theta (4–12Hz) rhythms. We concentrate on modeling in vitro experiments, but with an eye toward possible in vivo implications. Models of gamma and theta rhythms range from very detailed, biophysically realistic descriptions to abstract caricatures. At the most detailed levels, the cells are described by Hodgkin–Huxley-type equations, with many different cell types and large numbers of ionic currents and compartments (Traub et al., 2004). We use simpler biophysical models; all cells have a single compartment only, and the interneurons are restricted to two types: fast-spiking (FS) basket cells and oriens lacunosum-moleculare (O-LM) cells. Unlike Traub et al. (2004), we aim not so much at reproducing dynamics in great detail, but at clarifying the essential mechanisms underlying the production of the rhythms and their interactions (Kopell, 2005). In particular, we wish to highlight the dynamical as well as physiological mechanisms associated with rhythms, and to begin to classify them by mechanisms, not just frequencies. One theme in this chapter is the interaction of gamma and theta rhythms. To understand this interaction, it is necessary to describe the mechanisms of the gamma and theta rhythms separately before putting them together. A second theme is the use of mathematical tools to get a deeper understanding of the dynamics of rhythmic networks. We apply these ideas mainly to the question of how networks can produce