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

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 1 and a value greater than $2\phi -1$, where $\phi$ 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.

Figure 1

PRCs for leaky integrate and fire oscillator. $S_O=1.0$ and $\gamma =0.9$. These PRC shapes result when an instantaneous change in voltage of magnitude epsilon is applied to an oscillator with a monotonically increasing concave down membrane potential waveform between threshold crossings that result in a reset. Excitatory pulse coupling (dashed lines) and inhibitory pulse coupling (solid lines) each have two regions, a linear region where the slope is 1 due to the requirement to keep the phase between zero and 1, and a region in which the advance or delay increases with phase.

Figure 2

PRC protocol, cluster logic, and pulse-coupled firing patterns. (a) The PRC is measured (a1) by applying a pulse to a self-connected oscillator (a2) representing a cluster. (b1) In a two-cluster network each cluster receives an input from itself at a delay (which is inside the “black box” used for computing the PRC) and an input from an identical cluster. (b2) The logic is similar for larger clusters. (c) A fixed firing pattern sequence for $n=4$. A pulse is emitted when a “spike” occurs, and received after a delay $\delta$ as an input by the other clusters.

Figure 3

Graphical method. The phase of the last input $({\phi }_n^{*}$, indicated by the open circles) received by each cluster in an $n$-cluster mode with no delay falls at the intersection of these two functions described in the text.

Figure 4

Generalization to unequal time lags. For each time lag, there is a clear relationship between the four phases defined in relationship to this lag by the definitions of the stimulus, recovery, and intermediate intervals. However, unless the lags are equal, there is no fixed relationship between phases at which inputs are received in adjacent time lag intervals.

Figure 5

How delays destroy cluster modes. The intersection of the $g$ functions with the dashed lines labeled with $n=2$, 3, or 4 gives the phase of the last input for cluster modes. (a) With no delay and a linear PRC with slope 0.85, cluster modes 2, 3, and 4 are stable. (b) A normalized delay shown by the labeled bar shifts all lines to the left, destroying all cluster modes and leaving virtual (or ghost) repellers in their wake. This particular PRC with the illustrated delay guarantees that global synchrony is globally attracting; see text.