Counter-Intuitive Phenomena in Limit Cycle Systems with Applications to Neuroscience and Stochastic Dynamics
Time and place
1 PM on Tuesday, February 24th, 2015; NAC 6/327
Michael Schwemmer (Ohio State University)
Abstract
Limit cycle oscillators have been widely used to model a variety of physical systems. As such their responses to various forms of perturbations is of intense interest. In this talk, we show two different examples of how an external perturbation to a limit cycle system can lead to the emergence of a variety counter-intuitive behaviors.
The first example is motivated by the dynamics of neuronal oscillators. Neurons can have extensive spatial geometries, but they are often modeled as single-compartment objects that ignore the spatial anatomy of the cell. However, many neurons are not electrotonically compact, and single-compartment models cannot be expected to fully capture their behavior. To explore the effects of spatial properties, we model a neuron as a soma (cell body) with limit cycle dynamics attached to a single cylindrical passive dendrite, i.e. a 'ball-and-stick model'. The addition of the dendrite acts as an external perturbation to the somatic limit cycle dynamics. We find that the addition of the dendrite can either increase or decrease the frequency of oscillations. Using the theory of weak coupling, we identify key dynamical features that predict how the addition of the dendrite will modulate the oscillator's frequency.
In the second example, we focus on the effects of noisy input to limit cycle systems. Using simple planar limit cycle oscillators, we show that the addition of moderate noise can qualitatively alter the dynamics of the system. In particular, the system can rotate in the opposite direction of the deterministic limit cycle, cease oscillating altogether, or even display bistable switching. We explain these differences using standard techniques from stochastic calculus and recently developed stochastic phase reduction methods.
