Heterogeneity of time delays determines synchronization of coupled oscillators

Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.

Figure 1

Tract lengths and weights from 100 healthy subjects. Joint distribution (a), and histogram of weighted lengths for (b) intra- and (c) inter-hemisphere links.

Figure 2

Sketch of delay-imposed structure and connectivity matrices for oscillators in models A and B with link delays ${\tau }_1$ dark (blue) dashed lines and ${\tau }_2$ light (red) lines.

Figure 3

Critical couplings for incoherence for random bimodal-$\delta$ delays, Eq. (15), (a, b), and for structured models A, Eq. (16), (c) and B, Eq. (17), (d). (a) Lowest couplings $K<K_c$ (black surface) for steady, Eq. (18), stable, Eq. (19) solutions. (b) Intersection of plot (a) for ${\tau }_2=0.6$ (upper surface is thin blue and lower is dashed black line) and numerical results, Eq. (1), for incoherence (blue triangles), coherence (black squares) and bistability (grey shaded areas). Parameters: $\mu =2\pi ,\gamma =0.1$, and $p_1=0.5$.

Figure 4

Anti-phase clusters for model A with (a) 2, (b) 3 and (c) 9 equal populations. (a, b) Blue and red are for the first and second, and grey is for the overall population. (a) Magnitude of order parameters, theoretical (dashed blue, thick red and dotted grey), Eq. (13) and numerical results (blue squares, red circles and grey triangles), Eq. (1) for $N=100000$ oscillators. Inset: PDF of the phases. (b, c) Order parameters $z_m$. (c) Each of the three thicker arrows accounts for two identical mean-fields. Parameters: $K=2,\mu =2\pi ,\gamma =0.1$, (a) $\tau =[0.3,0.7]$, (b, c) $\tau =[0.15,0.55]$.

Figure 5

(a) Magnitudes of non-steady global order parameters, (b) mean field frequencies, and (c)–(e) adjusted frequencies for model B. (Green) arrows in (a, b) indicate the time points $t=[18.2,19.1,20]$ for the results in plots (c)–(e). Blue, red and grey correspond to the first, second, and the overall population. (a, b) Theoretical (dashed blue, thick red and dotted grey), Eq. (14), and numerical results (blue squares, red circles and grey triangles), Eq. (1). Parameters: $K=3,p_1=0.5,\mu =2\pi ,\gamma =0.1,\tau =[0.23,0.74],N=100000$.

Figure 6

Critical couplings for synchronization at EEG frequencies for all-to-all network with identical coupling strengths and different bimodal-$\delta$ spatial distributions of realistic brain delays, $\tau =[18$ ms, $42\mathrm{ms}],p_1=0.7$.