Within-burst synchrony changes for coupled elliptic bursters

We study the appearance of a novel phenomenon for linearly 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 end of burst occurs via a fold limit cycle bifurcation. We study synchronization behavior of two and three Bautin-type elliptic bursters for a linear direct coupling scheme. Burst synchronization is known to be prevalent behavior among such coupled bursters, while spike synchronization is more dependent on the details of the coupling. 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 spiking frequency.

💡 Research Summary

This paper investigates a previously unreported dynamical phenomenon that arises when identical elliptic bursters of Bautin type are coupled through a linear direct coupling scheme. Each isolated burster follows a normal‑form dynamics that exhibits a slow passage through a codimension‑two degenerate Andronov‑Hopf (Bautin) bifurcation: a subcritical Hopf bifurcation initiates repetitive spiking at the onset of a burst, while the burst terminates via a fold limit‑cycle bifurcation. The authors first confirm that the higher‑order nonlinear terms present in the normal form do not noticeably affect the trajectory of a single burster.

The core of the study examines two‑ and three‑burster networks. Using the standard linear coupling (\dot{x}i = f(x_i) + \kappa\sum{j}(x_j - x_i)), they explore a wide range of coupling strengths (\kappa). At the burst level, synchronization is robust: the slow variable of each unit aligns, causing the start and end times of bursts to coincide across the network. This agrees with earlier work showing that burst‑level synchrony is a generic outcome for coupled elliptic bursters.

In contrast, spike‑level synchrony proves to be highly sensitive to the details of the coupling and, crucially, to the higher‑order terms of the normal form. Although these terms leave the single‑burster dynamics essentially unchanged, they introduce subtle differences in the instantaneous spiking frequency of each unit when the system is coupled. As a burst progresses, the spiking frequency varies nonlinearly and possesses a turning point where the rate of change of frequency sharply reverses. Near this turning point the phase difference between coupled bursters can expand dramatically, producing a temporary loss of spike synchrony. The higher‑order nonlinear contribution then acts to reduce the phase gap, allowing the units to re‑synchronize later in the same burst. The authors refer to this transient loss and recovery of spike synchrony within a single burst as “within‑burst synchrony change.”

Numerical simulations confirm that this phenomenon occurs both for two coupled bursters and for three coupled bursters. In the two‑burster case, increasing (\kappa) generally strengthens spike synchrony, but for a specific interval of parameters the phase gap widens, leading to a clear desynchronization window inside the burst. For three bursters, symmetric coupling yields cluster states: two units may remain phase‑locked while the third lags, illustrating that network topology also influences the pattern of within‑burst synchrony changes.

The paper’s major contributions are: (1) identification of a new dynamical regime where spike synchrony can switch on and off within a single burst, (2) demonstration that higher‑order nonlinear terms—normally neglected in single‑unit analyses—are responsible for this regime, (3) linking the synchrony switches to a turning point in the spiking frequency curve, and (4) showing that the effect persists across small networks (N = 2, 3) and can generate cluster synchronization.

From a broader perspective, these findings have implications for neuroscience and engineered neural systems. In biological circuits, bursts often coexist with precise spike timing, and the ability of a network to alter spike synchrony mid‑burst could underlie complex rhythm generation, information coding, or pathological states such as tremor. In artificial neural networks, the results suggest that careful tuning of coupling strength and inclusion of appropriate higher‑order terms can be used to design networks with controllable synchrony patterns, potentially improving the performance of neuromorphic hardware that relies on burst‑based computation.

Overall, the study enriches the theoretical understanding of coupled elliptic bursters by revealing that within‑burst dynamics are not static but can undergo rapid synchrony transitions driven by subtle nonlinear interactions, opening new avenues for both analytical investigation and practical application.