Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

Abstract

The synchronization tendencies of networks of oscillators have been studied intensely. We assume a network of all-to-all pulse-coupled oscillators in which the effect of a pulse is independent of the number of oscillators that simultaneously emit a pulse and the normalized delay (the phase resetting) is a monotonically increasing function of oscillator phase with the slope everywhere less than one and a value greater than 2φ−1, where φ is the normalized phase. Order switching cannot occur; the only possible solutions are globally attracting synchrony and cluster solutions with a fixed firing order. For small conduction delays, we prove the former stable and all other possible attractors nonexistent due to the destabilizing discontinuity of the phase resetting at a phase of 0.

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity

“…They also studied the effects of conduction delays empirically and showed that with small conduction delays networks of Hodgkin-Huxley neurons such as the ones used in this study exhibited two or three clusters, but as the delay increased the number of possible clusters decreased to one, consistent with our results. The present study and our previous paper (Canavier and Tikidji-Hamburyan 2017) explain the role of conduction delays in reducing the number of clusters and are consistent with our previous experimental studies (Wang et al 2012) showing that conduction delays stabilize synchrony in neurons with reciprocal inhibitory synapses.…”

“…In an N-neuron network, many possible clusters exist, including three or more clusters or two clusters of unequal sizes. We have shown that for a monotonically increasing PRC that is essentially saturated no cluster states exist if the symmetrical two-cluster state does not exist (Canavier and Tikidji-Hamburyan 2017). In a state with more than two clusters, in general the phase at which an input is received by the cluster that fires immediately before it tends to be at a later phase than in a state with just two clusters, and therefore more likely to fall in the discontinuity (or region of negative slope for a weaker conductance; see Achuthan and Canavier 2009).…”

On the global synchronization of pulse-coupled oscillators interacting on chain and directed tree graphs

“…4, we can also see that the length of the shortest containing arc V c , which is widely used as a Lyapunov function in local synchronization analysis [24,29,32,34], is not appropriate for global PCO synchronization as it may not decrease monotonically. Along the same line, the firing order which is invariant in [4,28,42], and [32], is not constant in the considered dynamics as exemplified in Then we considered N = 10 PCOs on a directed tree graph, as illustrated in Fig. 1 (c Fig.…”

“…Recently, efforts have emerged to address global PCO synchronization from an arbitrary initial phase distribution. However, these results focus on special graphs, such as all-to-all graph [31,32,41,42], cycle graph [33], stronglyrooted graph [32], or master/slave graph [34]. Moreover, they rely on sufficiently large coupling strengths, which may not be desirable as large coupling strengths are detrimental to robustness to disturbances [4].…”

“…(Continuous) phase synchronization [31,32,41,42] [32, 33, 43 2 ] [34] This paper ment/maintenance. Furthermore, the chain graph has been regarded as the worst-case scenario for synchronization due to its minimum number of connections [44].…”