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Wave Patterns in Networks of Coupled Oscillators

Ding, Yujie (2023) Wave Patterns in Networks of Coupled Oscillators. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Recent advances in brain recording techniques have demonstrated that neuronal oscillations are not always synchronized, but rather, organized into spatio-temporal patterns such as traveling and rotating waves. This thesis is an investigation of wave patterns in a network of identically coupled phase oscillators. We demonstrate the existence and stability of traveling waves and rotating waves on a variety of domains with a combination of analytical and numerical methods, and also discuss the relationships between various types of coupling.
In Chapter 1, we bring in the concepts of neural oscillators as well as the phase reduction method.
In Chapter 2, we analyze a one-dimensional network of phase oscillators that are non- locally coupled via the phase response curve (PRC) and the Dirac delta function. The existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem.
We next extend this model from one-dimensional ring domains to two-dimensional annulus domains, and derive an integro-differential equation of the form commonly used to model two-dimensional neural fields. In Chapter 3, under the “weaker” weakly coupling setting, this network can be averaged into a diffusive phase coupling model.
In Chapter 4, we examine the existence, stability, and form of rigid rotating waves in a non-locally coupled phase model on the annulus. We show that as the hole in the annulus decreases, the waves lose stability. Through numerical simulations, we suggest that the bifurcation that occurs with the shrinking hole is a saddle-node infinite cycle and gives rise to so-called spiral chimeras.
In Chapter 5, rotating waves in a system of locally coupled phase oscillators on a N × N lattice grid are studied. We show that as N → ∞ that the dynamics can be understood by a Bessel equation on an annulus with inner radius proportional to 1/N. We find similar rotating wave patterns through simulations from both square lattice and hexagonal lattice.
A general discussion and an outlook of future work are provided in the final Chapter 6.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Ding, Yujieyud39@pitt.eduyud390000-0002-5645-7734
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairErmentrout,
Committee MemberRubin,
Committee MemberSwigon,
Committee MemberMugler,
Date: 25 January 2023
Date Type: Publication
Defense Date: 17 October 2022
Approval Date: 25 January 2023
Submission Date: 11 November 2022
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 148
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: traveling waves, rotating waves, spiral chimeras, nonlocal phase models
Related URLs:
Date Deposited: 25 Jan 2023 15:24
Last Modified: 16 Feb 2023 21:36


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