Connectionists: Western-Fields Seminar Series | Bard Ermentrout

[Lyle Muller]

The fourth talk in the 2021 Western-Fields Seminar Series in Networks, Random Graphs, and Neuroscience (http://www.fields.utoronto.ca/activities/21-22/western-fields) is next Thursday (8 July) at noon EDT.

Bard Ermentrout (U Pittsburgh, https://www.pitt.edu/~phase) will give a talk titled “Dynamics and patterns on graphs: Emergence of topological waves” (abstract below). Dr. Ermentrout is a leader in mathematical biology and applications of nonlinear dynamics to biological problems, from recurrent activity to oscillations and waves in a variety of neural systems.

This seminar series features monthly virtual talks from a diverse group of researchers across machine learning, physics, and graph theory. Upcoming speakers include Todd Coleman (University of California, San Diego), Krystal Guo (University of Amsterdam), Alexander Lubotzky (Hebrew University of Jerusalem), Frances Skinner (Krembil Institute), and more.

Registration link: https://zoom.us/meeting/register/tJYuf-GppzkjHt0W5HMDpME2UpUiE7ntO5JS

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Abstract: Rotating waves and similar non-synchronized patterns are often seen in large-scale local field potential (LFP) data.  It is now increasingly clear that these patterns and other waves are the norm.  In typical experiments, the LFP is band-pass filtered and the local phase is extracted from each electrode. The phase-gradients are then extracted with the result a spatial pattern of phases.  Here I will start with a graph with $N$ nodes and an adjacency matrix that determines which nodes are connected to each other. At each node, I assume that the dynamics is oscillatory and governed by a simple phase-model:

d theta_j/dt = w_j + sum_k A_jk H(theta_k-theta_j)

where w_j is the natural frequency, H(phi) is an interaction function, and A_jk is the adjacency matrix for the graph.  I will assume w_j=1, H(0)=0, H'(0)>0 so that synchrony (theta_j = t) is a stable solution.  In this talk I will ask what requirements there are on A such that there are other stable attractors.  These are typically patterns of phases that are organized about some local topological phase-singularity.  I will explore aspects of the graphs and methods to assign initial data to find these patterns.  I will first look at cubic graphs where we have fairly complete results for N<20 nodes.  I will also look at simple lattices (hexagonal and square), where we can construct rotating waves for N large.

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Lyle Muller
http://mullerlab.ca

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