Synchronization in networks with heterogeneous coupling delays

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

Figure 1

Complete synchronization and amplitude death in networks with heterogeneous delays.

Figure 2

(a) Structure of Eq. (21) with separation into master modes (${\mathcal{U}}_k$ contains only numbers), slave modes (${\mathcal{V}}_k$ contains only numbers), and intermediate modes. Only the squares and the diagonal stripes are nonempty. The stripes at the main diagonal are associated with Eq. (23) and determine the stability of the master and the slave modes, respectively. The intermediate modes are driven by the master modes and both can drive the slave modes. (b) Structure of Eq. (21) after additional decomposition of the intermediate modes. The two small squares at the main diagonal of the intermediate modes are associated with two blocks similar to Eq. (25), specifying the stability of the intermediate modes.

Figure 3

Stability charts for the equilibrium of Hodgkin-Huxley neurons with (a) all-to-all coupling and (b) general coupling. Thick (black) curves are associated with purely imaginary roots of the tangential mode 1. Thin dark (blue) and light (green) curves indicate purely imaginary roots associated with modes 2 and 4/5, respectively. Stable regions, where all characteristic roots have negative real part, are shaded. The dotted and dashed lines in panel (b) correspond to Figs. 4 and 4, respectively.

Figure 4

Real part of characteristic roots for general coupling with (a) homogeneous delays ${\tau }_1={\tau }_2$ and (b) heterogeneous delays ${\tau }_2={\tau }_1+4.8$ ms (b); cf. Fig. 3. Stable regions are shaded. Color code as in Fig. 3 (only the dominant roots corresponding to modes 1, 2, and 4/5 are shown).

Figure 5

Bifurcation diagrams for synchronized solutions of the Hodgkin-Huxley neurons for all-to-all coupling (left) and general coupling (right) and different values of the delay heterogeneity $\mathrm{\Delta }\tau ={\tau }_2-{\tau }_1$. The peak-to-peak voltage amplitude $|V_s|$ of the synchronized solutions are plotted as a function of the delay ${\tau }_1$. Solid thin green (thick red) curves represent stable (unstable) solutions with respect to tangential perturbations, while transversal instabilities are marked by dotted black curves. The parameters indicated by the dashed vertical lines in panels (a), (b), (e), and (f) are used in Fig. 6.

Figure 6

Voltages of the Hodgkin-Huxley neurons Eq. (35) for all-to-all coupling (a) and general coupling (b). We set ${\tau }_1=3.4$ ms while heterogeneity in the delays is introduced at $t=200$ ms by switching from $\mathrm{\Delta }\tau =0$ ms to $\mathrm{\Delta }\tau =2.4$ ms.