Jordan Snyder

Jordan Snyder

Assistant Professor of Mathematical Health and Disease Modeling

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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

Pages

Posts

Cell polarity model

less than 1 minute read

Published:

Together with Bjarke Nielsen, Ala Trusina, and Kim Sneppen, I am working on a model of organ growth with the aim of simulating the development of branched structures. As a basis I am taking the model used in this paper:

  • Nielsen BF, Nissen SB, Sneppen K, Mathiesen J, Trusina A. Model to Link Cell Shape and Polarity with Organogenesis. iScience. 2020 Feb 21;23(2):100830. doi: 10.1016/j.isci.2020.100830. Epub 2020 Jan 11.
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Sugar the dog

less than 1 minute read

Published:

My parents got a dog recently. Her name is Sugar and I love her very much. Read more

publications

Stability of entrainment of a continuum of coupled oscillators

Published in Chaos: An Interdisciplinary Journal of Nonlinear Science, 2017

Complex natural and engineered systems are ubiquitous, and their behavior is challenging to characterize and control. We examine the design of the entrainment process for an uncountably infinite collection of coupled phase oscillators that are all subject to the same periodic driving signal. In the absence of coupling, an appropriately designed input can result in each oscillator attaining the frequency of the driving signal, with a phase offset determined by its natural frequency. We consider a special case of interacting oscillators in which the coupling tends to destabilize the phase configuration to which the driving signal would send the collection in the absence of coupling. In this setting, we derive stability results that characterize the trade-off between the effects of driving and coupling, and compare these results to the well-known Kuramoto model of a collection of freerunning coupled oscillators. Read more

Recommended citation: Jordan Snyder, Anatoly Zlotnik, Aric Hagberg, "Stability of entrainment of a continuum of coupled oscillators." Chaos: An Interdisciplinary Journal of Nonlinear Science, 2017. http://aip.scitation.org/doi/10.1063/1.4994567

A Dynamic Niche Model for the Emergence and Evolution of Mutualistic Network Structures

Published in arXiv, 2018

Mutualistic interactions are vital constituents of ecological and socio-economic systems. Empirical studies have found that the patterns of reciprocal relations among the participants often shows the salient features of being simultaneously nested and modular. Whether and how these two structural properties of mutualistic networks can emerge out of a common mechanism however remains unclear. We propose a unified dynamic model based on the adaptation of niche relations that gives rise to both structural features. We apply Hutchinson's concept of niche interaction to networked cooperative species. Their niche relation evolves under the assumption of fitness maximization. Modularity and nestedness emerge concurrently through the accumulated local advantages in the structural and demographic distribution. A rich ensemble of key dynamical behaviors are unveiled in the dynamical framework. We demonstrate that mutualism can exhibit either a stabilizing or destabilizing effect on the evolved network, which undergoes a drastic transition with the overall competition level. Most strikingly, the adaptive network may exhibit a profound nature of history-dependency in response to environmental changes, allowing it to be found in alternative stable structures. The adaptive nature of niche interactions, as captured in our framework, can underlie a broad class of ecological relations and also socio-economic networks that engage in bipartite cooperation. Read more

Recommended citation: Weiran Cai, Jordan Snyder, Alan Hastings, Raissa D'Souza, "A Dynamic Niche Model for the Emergence and Evolution of Mutualistic Network Structures." arXiv, 2018. http://arxiv.org/abs/1812.03564

A Pulse-gated, Neural Implementation of the Backpropagation Algorithm

Published in Proceedings of the 7th Annual Neuro-inspired Computational Elements Workshop on - NICE '19, 2019

For some time, it has been thought that backpropagation of errors could not be implemented in biophysiologically realistic neural circuits. This belief was largely due to either 1) the need for symmetric replication of feedback and feedforward weights, 2) the need for differing forms of activation between forward and backward propagating sweeps, and 3) the need for a separate network for error gradient computation and storage, on the one hand, or 4) nonphysiological backpropagation through the forward propagating neurons themselves, on the other. In this paper, we present spiking neuron mechanisms for gating pulses to maintain short-term memories, controlling forward inference and backward error propagation, and coordinating learning of feedback and feedforward weights. These neural mechanisms are synthesized into a new backpropagation algorithm for neuromorphic circuits. Read more

Recommended citation: Andrew Sornborger, Louis Tao, Jordan Snyder, Anatoly Zlotnik, "A Pulse-gated, Neural Implementation of the Backpropagation Algorithm." Proceedings of the 7th Annual Neuro-inspired Computational Elements Workshop on - NICE '19, 2019. http://dl.acm.org/citation.cfm?doid=3320288.3320305

Thresholding normally distributed data creates complex networks

Published in Physical Review E, 2020

Network data sets are often constructed by some kind of thresholding procedure. The resulting networks frequently possess properties such as heavy-tailed degree distributions, clustering, large connected components, and short average shortest path lengths. These properties are considered typical of complex networks and appear in many contexts, prompting consideration of their universality. Here we introduce a simple model for correlated relational data and study the network ensemble obtained by thresholding it. We find that some, but not all, of the properties associated with complex networks can be seen after thresholding the correlated data, even though the underlying data are not "complex."In particular, we observe heavy-tailed degree distributions, a large numbers of triangles, and short path lengths, while we do not observe nonvanishing clustering or community structure. Read more

Recommended citation: George Cantwell, Yanchen Liu, Benjamin Maier, Alice Schwarze, Carlos Serv{\'{a}}n, Jordan Snyder, Guillaume St-Onge, "Thresholding normally distributed data creates complex networks." Physical Review E, 2020. https://doi.org/10.1103/PhysRevE.101.062302

Degree-targeted cascades in modular, degree-heterogeneous networks

Published in arXiv, 2020

Many large scale phenomena, such as rapid changes in public opinion and the outbreak of disease epidemics, can be fruitfully modeled as cascades of activation on networks. This provides understanding of how various connectivity patterns among agents can influence the eventual extent of a cascade. We consider cascading dynamics on modular, degree-heterogeneous networks, as such features are observed in many real-world networks, and consider specifically the impact of the seeding strategy. We derive an analytic set of equations for the system by introducing a reduced description that extends a method developed by Gleeson that lets us accurately capture different seeding strategies using only one dynamical variable per module, namely the conditional exposure probability. We establish that activating the highest-degree nodes rather than random selection is more effective at growing a cascade locally, while the ability of a cascade to fully activate other modules is determined by the extent of large-scale interconnection between modules and is independent of how seed nodes are selected. Read more

Recommended citation: Jordan Snyder, Weiran Cai, Raissa D'Souza, "Degree-targeted cascades in modular, degree-heterogeneous networks." arXiv, 2020. http://arxiv.org/abs/2004.09316

Thresholding normally distributed data creates complex networks

Published in Physical Review E, 2020

Network data sets are often constructed by some kind of thresholding procedure. The resulting networks frequently possess properties such as heavy-tailed degree distributions, clustering, large connected components and short average shortest path lengths. These properties are considered typical of complex networks and appear in many contexts, prompting consideration of their universality. Here we introduce a simple generative model for continuous valued relational data and study the network ensemble obtained by thresholding it. We find that some, but not all, of the properties associated with complex networks can be seen after thresholding, even though the underlying data is not "complex". In particular, we observe heavy-tailed degree distributions, large numbers of triangles, and short path lengths, while we do not observe non-vanishing clustering or community structure. Read more

Recommended citation: George Cantwell, Yanchen Liu, Benjamin Maier, Alice Schwarze, Carlos Serv{\'{a}}n, Jordan Snyder, Guillaume St-Onge, "Thresholding normally distributed data creates complex networks." Physical Review E, 2020. https://link.aps.org/doi/10.1103/PhysRevE.101.062302

Mutualistic networks emerging from adaptive niche-based interactions

Published in Nature Communications, 2020

Mutualistic networks are vital ecological and social systems shaped by adaptation and evolution. They involve bipartite cooperation via the exchange of goods or services between actors of different types. Empirical observations of mutualistic networks across genres and geographic conditions reveal correlated nested and modular patterns. Yet, the underlying mechanism for the network assembly remains unclear. We propose a niche-based adaptive mechanism where both nestedness and modularity emerge simultaneously as complementary facets of an optimal niche structure. Key dynamical properties are revealed at different timescales. Foremost, mutualism can either enhance or reduce the network stability, depending on competition intensity. Moreover, structural adaptations are asymmetric, exhibiting strong hysteresis in response to environmental change. Finally, at the evolutionary timescale we show that the adaptive mechanism plays a crucial role in preserving the distinctive patterns of mutualism under species invasions and extinctions. Read more

Recommended citation: Weiran Cai, Jordan Snyder, Alan Hastings, Raissa D'Souza, "Mutualistic networks emerging from adaptive niche-based interactions." Nature Communications, 2020. http://dx.doi.org/10.1038/s41467-020-19154-5

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

Published in Physica D: Nonlinear Phenomena, 2021

Obtaining coarse-grained models that accurately incorporate finite-size effects is an important open challenge in the study of complex, multi-scale 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 {\&}rarr; {\&}infin; 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. Read more

Recommended citation: Jordan Snyder, Jared Callaham, Steven Brunton, J. Kutz, "Data-driven stochastic modeling of coarse-grained dynamics with finite-size effects using Langevin regression." Physica D: Nonlinear Phenomena, 2021. http://dx.doi.org/10.1016/j.physd.2021.133004

talks

Computing Geometric Integrated Information

Published:

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). Read more

Entrainment of Coupled Oscillators

Published:

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). Read more

Modeling collective behavior in hierarchically-organized systems

Published:

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. Read more

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

Published:

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. Read more

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

Published:

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. Read more

Modeling and inference of collective behavior in complex system

Published:

“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. Read more

teaching

Scientific Computing and Data Science

Graduate course, RUC, Department of Science and Environment, 2023

Flipped-classroom format course, co-taught with Prof. Thomas Schrøder focused on giving an overview of major topics in scientific computing and data science. In-class activities followed Jupyter notebooks hosted here. Topics in the data science portion of the course follow the book Data-Driven Science and Engineering by Steve Brunton and J. Nathan Kutz, and make use of recorded lectures from Steve Brunton’s YouTube channel Read more