Breaking Synchrony by Heterogeneity in Complex Networks

For networks of pulse-coupled oscillators with complex connectivity, we demonstrate that in the presence of coupling heterogeneity precisely timed periodic firing patterns replace the state of global synchrony that exists in homogenous networks only. With increasing disorder, these firing patterns persist until a critical temporal extent is reached that is of the order of the interaction delay. For stronger disorder the periodic firing patterns cease to exist and only asynchronous, aperiodic states are observed. We derive self-consistency equations to predict the precise temporal structure of a pattern from a network of given connectivity and heterogeneity. Moreover, we show how to design heterogeneous coupling architectures to create an arbitrary prescribed pattern.

Figure 1

Heterogeneities in a random network ($N=100$, $p=0.4$, $b=2$, $\widehat{\epsilon }=-0.1$, $\tau =0.1$) induce desynchronization followed by a transition to an asynchronous, aperiodic state. (a) Relative phases (ten selected phases shown in black) for different disorder strengths $\Delta \epsilon$ (fixed $J$). (b) Increase of the extent $\Delta \varphi$ of the pattern shown in (a), until the pattern disappears at $\Delta {\varphi }_c\approx \tau$ (dashed line at $\Delta \varphi =\tau$). Inset: Onset of asynchronous state ($\Delta \varphi \approx 1$) for large disorder. (c),(d) Instantaneous network rate of (c) a periodic ($\Delta \epsilon =0.10$) and (d) an aperiodic ($\Delta \epsilon =0.15$) state in real time computed using a triangular sliding time window. Gray histograms on the right of (c) and (d) indicate rate distribution [excluding peak at zero network rate in (c)].

Figure 2

The critical extent of a pattern $\Delta {\varphi }_c$ as a function of the interaction delay $\tau$ ($N=100$, $b=2$, $\widehat{\epsilon }=-0.1$). Each data point is obtained for a different heterogeneity $J$ in fully connected networks (□) and random networks (○) of different connectivity $C$ with $p=0.2$. For the random network, data are averaged over 20 trials (dashed line: fit). Gray area indicates the theoretical second-order prediction of the transition zone between $\Delta {\varphi }_c^{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}}$ and $\Delta {\varphi }_c^{\mathrm{m}\mathrm{a}\mathrm{x}}$. Inset: Probability that a periodic pattern emerges from synchronous initial conditions for a given disorder $\Delta \epsilon$ (histogram of 100 random heterogeneities and connectivities, $p=0.2$, $\tau =0.1$).

Figure 3

Predicting firing patterns and designing coupling architectures to create a prescribed pattern ($b=2$, $\widehat{\epsilon }=-0.1$, $\tau =0.1$, predictions to first order). (a),(b) Comparison of prediction (gray lines) and simulation (markers) of selected phases of a pattern ($N=100$, $p=0.4$). (a) Phases shown period by period for $\Delta \epsilon =0.25$. Heterogeneity is instantaneously introduced at time $t=0$. The attractor is reached asymptotically. (b) Pattern for increasing disorder $\Delta \epsilon$ ($J$ fixed). Inset: standard deviation of prediction from simulation. (c) Coupling strengths (indicated by shading) in two different heterogeneous networks ($N=11$, left: $p=0.6$, right: $p=0.3$) designed to create a prescribed firing pattern (middle row, gray). Patterns in black are obtained by simulating these designed networks (initialized to the synchronous state).