Clustering behaviour in networks with time delayed all-to-all coupling

Networks of coupled oscillators arise in a variety of areas. Clustering is a type of oscillatory network behavior where elements of a network segregate into groups. Elements within a group oscillate synchronously, while elements in different groups oscillate with a fixed phase difference. In this thesis, we study networks of N identical oscillators with time delayed, global circulant coupling with two approaches. We first use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We then perform stability and bifurcation analysis to the original system of delay differential equations with symmetry. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to two specfi c examples: a network of FitzHugh-Nagumo neurons with diffusive coupling and a network of Morris-Lecar neurons with synaptic coupling. In the case studies, we show how time delays in the coupling between neurons can give rise to switching between different stable cluster solutions, coexistence of multiple stable cluster solutions and solutions with multiple frequencies

Nonlinear synchrony dynamics of neuronal bursters

We study the appearance of a novel phenomenon for coupled identical bursters: synchronized bursts where there are changes of spike synchrony within each burst. The examples we study are for normal form elliptic bursters where there is a periodic slow passage through a Bautin (codimension two degenerate Andronov-Hopf) bifurcation. This burster has a subcritical Andronov-Hopf bifurcation at the onset of repetitive spiking while the end of burst occurs via a fold limit cycle bifurcation. We study synchronization behavior of two Bautin-type elliptic bursters for a linear direct coupling scheme as well as demonstrating its presence in an approximation of gap-junction and synaptic coupling. We also find similar behaviour in system consisted of three and four Bautin-type elliptic bursters. We note that higher order terms in the normal form that do not affect the behavior of a single burster can be responsible for changes in synchrony pattern; more precisely, we find within-burst synchrony changes associated with a turning point in the spontaneous spiking frequency (frequency transition). We also find multiple synchrony changes in similar system by incorporating multiple frequency transitions. To explain the phenomenon we considered a burst-synchronized constrained model and a bifurcation analysis of the this reduced model shows the existence of the observed within-burst synchrony states. Within-burst synchrony change is also found in the system of mutually delaycoupled two Bautin-type elliptic bursters with a constant delay. The similar phenomenon is shown to exist in the mutually-coupled conductance-based Morris-Lecar neuronal system with an additional slow variable generating elliptic bursting. We also find within-burst synchrony change in linearly coupled FitzHugh-Rinzel 2 3 elliptic bursting system where the synchrony change occurs via a period doubling bifurcation. A bifurcation analysis of a burst-synchronized constrained system identifies the periodic doubling bifurcation in this case. We show emergence of spontaneous burst synchrony cluster in the system of three Hindmarsh-Rose square-wave bursters with nonlinear coupling. The system is found to change between the available cluster states depending on the stimulus. Lyapunov exponents of the burst synchrony states are computed from the corresponding variational system to probe the stability of the states. Numerical simulation also shows existence of burst synchrony cluster in the larger network of such system.Exeter Research Scholarship