Research Coop: Collective dynamics in systems with multiple timescales

In the Spring and Summer of 2021, Matthew Mizuhara and I worked with two undergraduate students on projects concerning dynamics in large systems of coupled oscillators. Andrew Phillips and Justin Gliksman, undergraduate physics and math majors at Drexel University, investigated dynamics in population of oscillators near a saddle-node bifurcation.

The goal of this work was to identify new phenomena in collective dynamics due to proximity of the oscillators to the saddle-node bifurcations. To this end, students conducted numerical experiments with coupled systems of various oscillators including

Morris-Lecar \[\dot{V} = I - g_L(V-E_L) - g_{Na}m_{\infty}(V)(V-E_{Na}) - g_kn(V-E_k)\] \[\dot{n} = \phi(n_{\infty}(V) - n)\tau(V)\] \[m_\infty(v) = 0.5(1 + \tanh(\frac{v - \nu_1}{\nu_2}))\] \[n_\infty(v) = 0.5(1 + \tanh(\frac{v - \nu_3}{\nu_4}))\] \[\tau(v) = (\cosh(\frac{v - \nu_3}{2\nu_4}))^{-1}\]

and Stewart-Landau type oscillators \[ \dot{x} = x(1 - x^2 - y^2) - y(x - b)\] \[\dot{y} = y(1 - x^2 - y^2) + x(x - b).\]

Numerical experimenets revealed that in such systems coherence within the population may undergo robust oscillations, as shown in the following simulations.



Student Presentations

In the course of this coop, students prepared multiple presentation on the material that they learned for this project.

• Justin Gliksman, Type 1 Membranes, Phase Resetting Curves, and Synchrony
• Justin Gliksman, Phase Amplitude Descriptions of neural Oscillator Models
• Andrew Phillips, Selected Readings
• Andrew Phillips, When Nonlocal Coupling Becomes Stronger: Patched Synchrony of Multichimera states
• Andrew Phillips, Phase Reduction fo Relaxation Oscillators using malkin's Theorem
• Andrew Phillips, Robustness of Chimera States for coupled for Coupled Fitzhugh Nagumo Oscillators
• Andrew Phillips, Moving Bumps in Theta Neuron Networks
• Andrew Phillips, Derivation of Neural field Model

Software

Software developed in the course of this activity is avaibale through the following

• Github Repository

Future Work

In the future, we plan to analyze the origins of the oscillations of the coherence in such systems.

Related papers

• When Nonlocal Coupling Between Oscillators Becomes Stronger: Patched Synchrony of Multi-Chimera States
• Robustness of chimera states for coupled Fitzhugh Nagumo Oscillators
• Moving Bumps in Theta Neuron Networks
• Emergence of second coherent regions for breathing chimera states
• Chimerapedia: Coherence-incoherence patterns in one, two, and three dimensions
• Derivation of neural field model from a network of theta neurons
• Stability of clusters in the second-order Kuramoto model on random graphs
• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Ott-Antonsen ansatz for the D-dimensional Kuramoto model: a constructive approach.
• Phase Equations for Relaxation Oscillators
• Clustered chimera states in systems fo type-I excitability