Mathematical Biology Seminar|
Department of Mathematics, University of Utah
3:05PM, Wednesday September 26, 2012
"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.