Mathematical Biology Seminar Alla Borisyuk Department of Mathematics, University of Utah 3:05PM, Wednesday September 26, 2012 LCB 225 "Periodically driven noisy neuronal models: a spectral approach" Neurons are often driven by periodic or periodically modulated inputs. The response of the neurons is often periodic and phase-locked to the stimulus. However, this is not always the case. It is well-known that even simple deterministic systems can exhibit quasiperiodic behavior, with no clear relationship between the phases of the stimulus and response. Moreover, the biological situation is further complicated by presence of noise in the periodic inputs and intrinsic cellular properties, e.g. firing rate adaptation. In an effort to develop a mathematical theory applicable to the above biologically-motivated project on phase-locking and related phenomena, we have developed a spectral approach to stochastic circle maps. A stochastic circle map is defined as a Markov chain on the circle. This abstract class of objects includes a wide range of models for firing times of periodically forced noisy neuronal models. We analyze path-wise dynamic properties of the Markov chain, such as stochastic periodicity (or phase locking) and stochastic quasiperiodicity, and show how these properties are read off of the geometry of the spectrum of the transition operator.