Symmetry Breaking in a Delay Differential Equation Modelling Auditory Streaming

Abstract

This thesis develops new methods for bifurcation analysis of delay-differential equations (DDEs). These methods can be used to find and track symmetry-breaking bifurcations of periodic orbits in DDEs with discrete symmetry. The method development is motivated by a modelling problem posed by Ferrario and Rankin (2021) who studied perceptual bistability of the auditory system. The Ferrario and Rankin (2021) model describes two coupled identical neural populations with delayed cross-inhibition coupling, such that the model is a four-dimensional system with periodic forcing. The model has a reflection symmetry, and the breaking of this symmetry corresponds to a switch between two distinct auditory perceptions. The thesis starts with an overview of the general concepts of DDEs and bifurcation theory. We then develop and apply specialized techniques for tracking symmetric solutions and symmetry-breaking bifurcations, extending existing methods for tracking steady-state solutions and periodic orbits by enforcing symmetry on a fixed-point sub- space of equivariant systems. These methods are implemented in DDE-Biftool. We use our symmetry-extended defining systems to perform a bifurcation analysis on the periodically forced auditory streaming model proposed by Ferrario and Rankin (2021) (after reformulating it as an autonomous DDE). By studying the dependence of the perception boundary on the choice of threshold, our bifurcation analysis precisely determines the perception switching boundaries in the space of two experimentally relevant input parameters, the tone frequency difference and the presentation rate. These boundaries appear to approximate the symmetry-breaking solutions, identified by analysing non-symmetric periodic solutions in which the maximum activity of one neural population touches the threshold during the active-tone interval of the other neural population. We find that the dynamical bistability requires symmetry-breaking, but symmetry-breaking is not sufficient for perceptual bistability. We extend the study by Ferrario and Rankin (2021) by varying additional parameters such as the tone duration, the coupling delay, and the internal time scale of the excitatory neural population. Further, we find that increasing the delay leads to new phenomena such as period doubling and torus bifurcations which were not observed in the original study of Ferrario and Rankin (2021), but which may be present in experiments. Additionally, we find that reducing the internal time scale for the excitatory neural population results in more realistic neural pulses, but it reduces agreement with the experimental data in parameter space suggesting that other system parameters need to be adjusted.