October 09, 2020, Zoom

Our goal is to understand the effect of stochasticity on coupled oscillators' dynamics, and in particular, their synchronization behavior. First, we consider a single noisy oscillator and discuss its phase reduction. Using the first and second-order Phase Response Curves (PRCs), we derive a reduced-order stochastic differential equation representing the phase evolution. We approximate the first and second moments of the oscillators' time period in terms of functions of the PRCs to understand the effect of noise on these periods. Then, we consider a network of noisy oscillators and assume that two state-dependent noise sources drive each oscillator: (1) a common noise, and (2) noise through interactions with other oscillators. Leveraging the PRCs, we derive a reduced system of coupled phase equations. Then, we provide sufficient conditions, based on the PRCs and the state-dependent diffusion matrics, that foster synchronization. We will illustrate the role of each source of the noise in a numerical example.

Zahra Aminzare is an Assistant Professor at the Department of Mathematics, University of Iowa. Before that, she was a Postdoctoral Researcher in the Program in Applied and Computational Mathematics at Princeton University from 2015-2018. She received her Ph. D. in Mathematics from Rutgers University in 2015. Zahra is interested in employing and developing mathematical models, dynamical systems techniques, and numerical simulations to better understand networks' collective behavior. Part of her work is motivated by neuroscience applications, such as understanding central pattern generator networks' activity by studying the underlying mechanisms of gait patterns in insects and transition between the gaits. She is also interested in understanding the collective behavior of bacteria, such as E. coli, in response to external signals and analyzing their decision-making dynamics in response to multiple external signals.