Jordan Snyder

Jordan Snyder

Assistant Professor of Mathematical Health and Disease Modeling

Talks and presentations

Modeling and inference of collective behavior in complex system

May 10, 2023

Talk, IMFUFA Seminar, RUC

“Collective behavior” is a term that evokes many of most enigmatic and fascinating phenomena in the world: animal flocking, information spreading and social dynamics, coherent structures in fluid flows, traffic jams, cascading failures in electric power or other infrastructure systems; and so on. There is a rich tradition of building models that help us characterize, predict, or otherwise understand such systems. My goal is to build mathematical tools to take data from complex systems and deliver a mathematical description of the collective behavior it undergoes. To do this I use toy models, specifically coupled oscillators (the Kuramoto model). I will give two examples of such work and discuss ongoing work to incorporate a further level of model complexity, namely memory.

Data-driven stochastic modeling of coarse-grained dynamics with finite-size effects using Langevin regression

May 25, 2021

Talk, SIAM Conference on Applications of Dynamical Systems (DS21), Virtual

Obtaining physically meaningful coarse-grained models that incorporate finite-size effects is an important open challenge in the study of complex systems. We apply Langevin Regression, a recently developed method for finding stochastic differential equation (SDE) descriptions of realistically-sampled time series data, to understand finite-size effects in the Kuramoto model of coupled oscillators. We find that across the entire bifurcation diagram, the dynamics of the Kuramoto order parameter are statistically consistent with an SDE whose drift term has the form predicted by the Ott-Antonsen ansatz in the N→∞ limit. We find that the diffusion term is nearly independent of the bifurcation parameter, and has a magnitude decaying as N−1/2, consistent with the Central Limit Theorem. This shows that the diverging fluctuations of the order parameter near the critical point are driven by a bifurcation in the underlying drift term, rather than increased stochastic forcing.

Coarse-graining for coupled oscillators: a case study in discovering low-dimensional dynamics

October 04, 2019

Talk, CSU Northridge Applied Math Seminar, Northridge, CA

Numerical simulations form a backbone of modern science. We investigate the following question: given a simulation of some dynamical process, does there exist a good lower-dimensional representation? If so, finding such a representation may offer both computational speedup and fundamental insight into the dynamics of interest. To approach this question in the abstract, we infer coarse-grained equations of motion that describe a heterogeneous population of oscillators with a modular coupling structure. We choose this system because it is known to exhibit a transition from high- to low-dimensional behavior, and that low-dimensional behavior is well-described by equations of a known form. We conclude by exploring ways to move forward by systematically discarding several of the simplifying assumptions at play.

Modeling collective behavior in hierarchically-organized systems

August 17, 2017

Talk, Center for Nonlinear Studies, Los Alamos National Lab, Los Alamos, New Mexico

Collective behavior is the organizing principle that has made existence anything more than a homogeneous expanse of mass and energy. Understanding how collective behavior emerges and operates is not only fascinating as a purely intellectual exercise, but compelling from an engineering perspective as well. In this talk I will describe how hierarchical structures arise naturally in systems that exhibit collective behavior, and describe two mathematical modeling projects aimed at giving a quantitative understanding of the building blocks of collective behavior in hierarchically organized systems.

Entrainment of Coupled Oscillators

October 12, 2016

Talk, UC Davis, Student-run math/applied math seminar, Davis, California

Complex systems are ubiquitous, and often difficult to control. As a toy model for the control of a complex system, we take a system of coupled phase oscillators, all subject to the same periodic driving signal. It was shown by Anatoly Zlotnik and collaborators that in the absence of coupling, this can result in each oscillator attaining the frequency of the driving signal, with a phase offset determined by the oscillator’s natural frequency. We consider a special case in which the coupling tends to destabilize the phase configuration to which the driving signal would send the oscillators in the absence of coupling. In this setting we derive stability estimates that capture the trade-off between driving and coupling, and compare these results to the unforced version (i.e. the standard Kuramoto model).

Computing Geometric Integrated Information

June 01, 2016

Talk, UC Davis, Student-run math/applied math seminar, Davis, California

For decades, scientists across many disciplines, from biology to physics to mathematics, have been fascinated by emergent phenomena - complex behavior that arises from the interaction of many simple units. One approach to uncovering the underlying mechanisms behind emergence comes from information theory, which aims to quantitatively describe coordination between units in terms of production, transmission, and processing of information. In this talk, I will present a unified geometric interpretation of several different information measures intended to quantify causally-relevant information flow among components of a stochastic process, and calculate them on multi-layered primate social network data recorded by the McCowan lab at the California National Primate Research Center (CNPRC).