Resource-Constrained Synchrony: Kuramoto Oscillators Competing for Shared Resources

Many systems of biological interest exhibit oscillatory behavior, from the beating of a heart to the firing of neurons to the flashing of fireflies. Further, these oscillatory agents are rarely isolated from one another, and so may interact with one another. In the presence of such interactions, one possible outcome is synchronization of the oscillatory motions. Such synchrony may be observed in the simultaneous flashing of a great many fireflies, or the simultaneous firing of many neurons during an epileptic seizure. A classic model that captures this synchronization is the Kuramoto model. However, the Kuramoto model is a toy model, and thus much work has been directed to extending the model by introducing additional dynamics. In the dissertation, we will present two extensions of the Kuramoto model that make it more appropriate to the study or neural systems. The first extension will add a resource dependence to the Kuramoto dynamics, making the internal dynamics of the oscillators more complex, and thereby introducing novel phases into the Kuramoto phase diagram (Chapter 2). The next extension will allow the oscillators to compete for a shared supply of resources, creating a secondary avenue of communication between the oscillators (Chapter 4). This additional communication pathway will generate correlations in behavior, which may have some relevance for the differences observed between functional and structural connectivity measures in the brain. These two studies serve to elucidate some interesting results on the dynamics of Kuramoto oscillators competing for shared resources, and so serve as my primary contribution to the study of the physics of synchronizable systems. Further, as a scientist-educator, I am also interested in and committed to the education of young physicists, and so I have pursued a separate line of inquiry that studies the learning of students in a cross-disciplinary network-neuroscience course using the tool of concept networks (Chapter 6). We will find that student-drawn concept networks are a useful tool in studying the learning process at a high level, but that more thought needs to be put toward optimizing the collection task in order to bring out the full power of this tool. Collectively, these three studies --- two in the physics of dynamical systems and one in education --- have enabled me to develop in my role as a scientist-educator

Synchronization of coupled Kuramoto oscillators competing for resources

Populations of oscillators are present throughout nature. Very often synchronization is observed in such populations if they are allowed to interact. A paradigmatic model for the study of such phenomena has been the Kuramoto model. However, considering real oscillations are rarely isochronous as a function of energy, it is natural to extend the model by allowing the natural frequencies to vary as a function of some dynamical resource supply. Beyond just accounting for a dynamical supply of resources, however, competition over a \emph{shared} resource supply is important in a variety of biological systems. In neuronal systems, for example, resource competition enables the study of neural activity via fMRI. It is reasonable to expect that this dynamical resource allocation should have consequences for the synchronization behavior of the brain. This paper presents a modified Kuramoto dynamics which includes additional dynamical terms that provide a relatively simple model of resource competition among populations of Kuramoto oscillators. We design a mutlilayer system which highlights the impact of the competition dynamics, and we show that in this designed system, correlations can arise between the synchronization states of two populations of oscillators which share no phase-coupling edges. These correlations are interesting in light of the often observed variance between functional and structural connectivity measures in real systems. The model presented here then suggests that some of the observed discrepancy may be explained by the way in which the brain dynamically allocates resources to different regions according to demand. If true, models such as this one provide a theoretical framework for analyzing the differences between structural and functional measures, and possibly implicate dynamical resource allocation as an integral part of the neural computation process.Comment: 12 pages, 2 figure