Nonlinear synchrony dynamics of neuronal bursters

Abstract

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.