Coexistence of Regular and Irregular Dynamics in Complex Networks of Pulse-Coupled Oscillators

For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multioperator problem exactly and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying regular dynamics, coexists with a balanced state exhibiting irregular dynamics. External signals may switch the network between qualitatively distinct states.

Figure 1

Coexistence of (a) synchronous and (b) irregular dynamics in a random network ($N=400$, $p=0.2$, $I=4.0,$ $ϵ=16.0$, and $\tau =0.035$). (a),(b): Trajectories of the potential $U({\phi }_i)$ of three oscillators [angular bars: time scale (horizontal) $\Delta t=8$; potential scales (vertical) (a) $\Delta U=8$, (b) $\Delta U=2$; spikes of height $\Delta U=1$ added at firing times]. (c),(d): Distributions (c) $p_{\nu }$ of rates and (d) $p_{\mathrm{C}\mathrm{V}}$ of the coefficients of variation, displayed for the irregular (dark gray) and synchronous (light gray) dynamics.

Figure 2

Switching between synchronous and irregular dynamics ($N=400$, $p=0.2$, $I=4.0$, $ϵ=16.0$, and $\tau =0.14$). Firing times of five oscillators are shown in a time window $\Delta t=240$. Vertical dashed lines mark external perturbations: (i) large excitatory pulses lead to a synchronous state; (ii) a small random perturbation ($|\Delta {\phi }_i|\le 0.18$) is restored; (iii) a sufficiently large random perturbation ($|\Delta {\phi }_i|\le 0.36$) leads to an irregular state. Bottom: Time evolution of the spread of the spike times after perturbation (ii), total length $\Delta t=0.25$ each.