Forced networks of oscillating neurons are ubiquitous in the nervous system. Although forced unitary oscillators have been heavily studied, there are very few studies of the effects of periodic forcing on networks of oscillators with multiple internal degrees of freedom. Understanding such systems is an essential step in understanding how the brain makes use of its rhythmic dynamics to communicate signals among regions. The networks in this project have an almost universal property of networks of neurons: there is exponentially decaying inhibition that feeds back to the excitatory cells of the network. In these projects, the decaying nature of the inhibition is important for the emergence of lower-dimensional dynamics and the ability to produce and analyze simplified models of network phenomena. The initial three subprojects concern noise, multiple periodic inputs, and intrinsic currents providing extra timescales that interact with periodic forcing. The work will relate the properties of network interaction, such as the decaying inhibition, to geometric ideas of invariant manifold theory. The aim is to produce a level of mathematical clarity that is not generally a part of initial modeling work, and to expand the set of tools and concepts available for further study of forced networks. The project focuses on networks producing gamma (30-80 Hz) oscillations and beta (12-30 Hz) oscillations, with physiological descriptions of the neurons appropriate to those rhythms.
The dynamics of the nervous system are central to cognitive function, but how the brain makes use of its dynamics is barely beginning to be understood. Rhythms of the nervous system have long been known to be highly associated with cognitive processes including sensory coding, attention, learning, memory, and motor planning. The study of such rhythms, and their use in brain communication, is an excellent bridge between physiology and function. This project deals with the response of networks of neurons to periodic and other input, as would be seen from signals arriving from other parts of the brain. The purpose of the project is to understand how brain networks process their temporally and spatially coded inputs. This work is being done in the context of a broad initiative to study the origin and significance of brain rhythms. Kopell is the founder and current head of the Cognitive Rhythms Collaborative, a group of over two dozen Boston-based senior investigators interested in the physiological and dynamical mechanisms of neural activity, their importance in cognition, and their pathology in neurological diseases including schizophrenia, Parkinson's disease and epilepsy, as well as their changes in anesthesia; it includes experimentalists working in vivo and in vitro, mathematicians, statisticians and medical personnel, who are already incorporating ideas from the mathematical research into clinical practice.
