Oscillator models are fundamental in modeling systems with rhythmic behavior, such as locomotion. (First-order) Phase Response Curves (PRCs) provide essential information about how these oscillator models perform in a neighborhood of a stable limit cycle and facilitate a reduction of a high dimensional model to a one-dimensional phase model. Furthermore, when multiple oscillator models interact with each other, such one-dimensional reduced models enable the development of coupled oscillator models that use only the phase information and relative timing of their limit cycles. In the first part of this talk, I review this technique and explain how it helps to understand the generation of various locomotion gait patterns in insects and transitions between the gaits. In the second part of the talk, I discuss the generalization of the phase reduction technique to noisy oscillator models. I introduce the notion of “second-order” PRCs and leverage the first- and second-order PRCs to derive a stochastic phase equation that describes phase evolution in noisy oscillator models. I describe the computation of the distribution and moments of noisy oscillators’ time periods and illustrate the theoretical results on a noisy Hopf bifurcation normal form, a noisy Van der Pol oscillator, and a noisy bursting neuron model. This is joint work with Philip Holmes (Princeton University) and Vaibhav Srivastava (Michigan State University).
Zahra Aminzare received her B.S. degree in Mathematics from the Sharif University of Technology in Iran, and the Ph. D. in Mathematics from Rutgers University, in 2015. She was a Postdoctoral Researcher in the Program in Applied and Computational Mathematics at Princeton University from 2015-2018. She joined the Department of Mathematics at the University of Iowa, in 2018. Zahra is interested in employing and developing mathematical models, dynamical systems techniques, and numerical simulations to better understand the collective behavior of coupled cell networks. The main goal of her research is to study the effect of the intrinsic dynamics of network elements and their coupling interactions on the emergence of various patterns in networks. Part of her work is motivated by neuroscience applications, such as understanding the activity of central pattern generator networks in insects 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 the dynamics of their decision-making in response to multiple external signals.