Abstract. 
Neurons are often driven by (noisy) periodic or periodically modulated
inputs. In many such cases neuronal firing can be characterized by a
stochastic phase response map (SPRM) that maps phase of the current
spike into the phase of the subsequent spike. More generally, SPRMs
represent Markov chains on a circle. In our spectral approach to
studying such maps, we analyze path-wise dynamic properties of the
Markov chain, such as stochastic periodicity (or phaselocking) and
stochastic quasiperiodicity, and show how these properties are read
off  of the geometry of the spectrum of the transition operator. I
will also discuss how SPRMs can be computed for some neuronal models,
their relationship with phase response curves, and how they are
affected by changes in the ionic channels.