## Thesis Format

Integrated Article

## Degree

Master of Science

## Program

Neuroscience

## Collaborative Specialization

Machine Learning in Health and Biomedical Sciences

## Supervisor

Muller, L.

## Abstract

A fundamental problem in network science is to establish connections between the structure of a network and the resulting non-linear dynamics on the network. Here, we employ a recently introduced complex-valued approach to the finite Kuramoto Model, which is a canonical model used throughout neuroscience to study neural synchronization. This analytical approach connects network structure to emergent dynamics through the eigensystem of a composed matrix incorporating the network adjacency matrix, coupling strength and phase lag. We apply this approach in the non-asymptotic limit to generate (1) a mechanistic theory of finite chimera dynamics and (2) a method for predicting relaxation time to synchrony. These findings highlight a rare example where we can link network structure to the emergent non-linear dynamics analytically, bolstering the utility of novel approaches in network theory that include operator-based methods and extensions to higher order number fields.

## Summary for Lay Audience

Network dynamics refers to the study of networks of inter-connected units, or nodes that change their state across time. From this perspective, the brain can be modelled as a network of a finite number of interconnected nodes, where each node represents a distinct region of the brain, and its dynamics represent the collective neural activity at the given region. Patterns of synchronous and asynchronous neural activity have been connected to a variety of neurological functions, such as memory, attention, and information processing. The presence of simultaneous synchrony and asynchrony within regions of a given network is known as a chimera state. In such networks with a finite number of nodes, a mechanistic theory that can predict and explain the emergence of such chimera states is still yet to be seen. In addition, it also remains an open problem for finite networks to develop mathematical techniques that can connect the underlying structure of a network to predict features of its emergent dynamics, such as the time it takes for the network to synchronize, known as the relaxation time. These problems are further obscured because real-world network dynamics are highly non-linear, which usually precludes explicit and analytical mathematical approaches. We study these problems under the broader mathematical lens of the finite Kuramoto Model, a central model at the intersection of computational neuroscience and network theory. In this case, we have developed an analytical approach passing through the complex numbers which provides a novel link between the structural features of a Kuramoto network to the emergent collective dynamics. This link is phrased in terms of a set of functions called the eigensystem of the network structure and parameters extended to the complex numbers. In Chapter 1, we leverage this to predict and explain the emergence of chimera dynamics for finite Kuramoto networks. In Chapter 2, our approach leverages functions of the eigensystem to provide a prediction method for relaxation time to synchrony. These findings bolster the utility of novel spectral approaches to finite network dynamics, and may provide future insight into prediction and modelling of complex network systems in neuroscience and beyond.

## Recommended Citation

Graham, James W., "Oscillator Network Spectra, Chimeras, and Transient Dynamics in the Non-Asymptotic Limit" (2024). *Electronic Thesis and Dissertation Repository*. 10156.

https://ir.lib.uwo.ca/etd/10156

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

#### Included in

Computational Neuroscience Commons, Dynamical Systems Commons, Non-linear Dynamics Commons, Ordinary Differential Equations and Applied Dynamics Commons