Role of long cycles in excitable dynamics on graphs

Topological cycles in excitable networks can play an important role in maintaining the network activity. When properly activated, cycles act as dynamic pacemakers, sustaining the activity of the whole network. Most previous research has focused on the contributions of short cycles to network dynamics. Here, we identify the specific cycles that are used during different runs of activation in sparse random graphs, as a basis of characterizing the contribution of cycles of any length. Both simulation and a refined mean-field approach evidence a decrease in the cycle usage when the cycle length increases, reflecting a trade-off between long time for recovery after excitation and low vulnerability to out-of-phase external excitations. In spite of this statistical observation, we find that the successful usage of long cycles, though rare, has important functional consequences for sustaining network activity: The average cycle length is the main feature of the cycle length distribution that affects the average lifetime of activity in the network. Particularly, use of long, rather than short, cycles correlates with higher lifetime, and cutting shortcuts in long cycles tends to increase the average lifetime of the activity. Our findings, thus, emphasize the essential, previously underrated role of long cycles in sustaining network activity. On a more general level, the findings underline the importance of network topology, particularly cycle structure, for self-sustained network dynamics.

Figure 1

Examples of ER random graphs used in the study, after pruning their dangling ends. These examples are ordered according to the average cycle length, from values close to 11 up to 20, from left to right and top to bottom. Note that short cycles can be embedded in longer cycles.

Figure 2

Cycle usage varies qualitatively with the recovery probability. (a) and (c) Cycle usage as a function of time steps, for an ER random network with 50 nodes and 60 links, and two different recovery probabilities. Cycles are sorted by increasing length (cycle index on the left vertical axis) and their length is encoded in color (see the color bar). The initial conditions are the same for both simulations (see text) and the first 10 steps are discarded. The recovery probability is $p=0.9$ in (a) and (b), and $p=0.4$ in (c) and (d). (b) and (d) Average activity (total number of nodes in the active state at a given time divided the total number of nodes) as a function of time steps for the same network and initial conditions as in (a) and (c). To present a clear image we only draw the shortest 200 cycles (the other, longer, cycles were not used in complete turns in this case).

Figure 3

Trade-off between cycle recoverability and vulnerability to external excitations. The figure displays the average number of complete excitation turns achieved by $L$-cycles, Eq. (1) as a function of their length $L$. The average is performed over simulation runs (200 different initial conditions), then over $L$-cycles, then over 100 network realizations.

Figure 4

Trade-off between average lifetime decrease (at low $p$) and disruption of short cycles (at high $p$). The figure displays the average number of $L$-cycles used per simulation run as a function of the length $L$, for various recovery probabilities $p$. The average is performed over simulation runs (200 different initial conditions), then over $L$-cycles, then over 100 network realizations. Error bars correspond to the dispersion observed across network realizations.

Figure 5

Cycle usage observed in the simulation. The figure displays Eq. (2), as a function of the cycle length, for various recovery probabilities p.

Figure 6

Cycle usage predicted by the mean-field approach. The figure displays the mean-field prediction for $\gamma {(L,p)}^L$, Eq. (5), as a function of the cycle length, for various recovery probabilities $p$.

Figure 7

Lifetime of the excitable dynamics varies with the average length of a topological cycle. The figure displays the average lifetime of the network activity for different values of the recovery probability $p$, as a function of the average cycle length in the graph. The average lifetime was computed running 200 different initial conditions for each network, and we here present for each value of $p$ a scatter plot for 200 network realizations.

Figure 8

Long excitation lifetime correlates with usage of long cycles. The figure displays the average maximum length used for each network realization and the corresponding average lifetime (average over runs for both the maximum length and lifetime, scatter plot over 100 network realizations). The recovery probability is $p=0.6$.

Figure 9

Usage of long cycles is functionally essential. The figure displays the sum of the average time $L{\langle N\left(L\right)\rangle }_{\mathrm{runs}}$ spent in cycling excitation over cycles of length $L$ smaller than 7 ($S_1$, light gray dots) and equal or greater than 7 ($S_2$, black squares), and the corresponding average lifetime (average over runs, scatter plot for 100 network realizations). The recovery probability is $p=0.6$.

Figure 10

Shortcuts hinder or reduce long cycle usage. Average number of complete excitation turns (over runs) achieved by a cycle as a function of the number of shortcuts present in the cycle (scatter plot for 100 network realizations and all their cycles). The recovery probability is $p=0.6$.

Figure 11

Removal of shortcuts increases the lifetime. Histogram (over 100 network realizations and over 2000 initial conditions) of the percentage of edges among shortcuts (light gray) or nonshortcuts (black), whose removal increases the average lifetime.

Figure 12

Dynamical features of excitable networks and their topological determinants. Representation of (a) edge usage, with edge darkness increasing with its usage, (b) shortcuts of cycles (green edges), and (c) edges whose removal produce an increase of the average lifetime when rerunning the simulation (in red).

Figure 13

Counting cycling excitation. Schematic representation of one (a) and two (b) excitations traveling through a cycle. Excited nodes (E) are plotted in red, susceptible nodes (S) in yellow, and refractory nodes (R) in green. The arrows indicate the direction of excitation propagation.

Figure 14

Enhanced refractory period displays the same recoverability/vulnerability trade-off. Average number of turns of $L$-cycles for the original model $M_1$ and the modified models $M_2$, $M_3$, and $M_4$, for a recovery probability $p=0.9$. The number of turns for each network was computed using 200 initial conditions, and the final average was computed over 100 different network realizations [see Eq. (1)].

Figure 15

Refractory time dispersion correlates with the number of used cycles. Average number of cycles used per run versus the average refractory time, for the original model (black dots) at increasing average refractory time $1/p$, hence increasing standard deviation $\sigma =\sqrt{1-p}/p$, and the modified model (red squares) for $n=2,3,4$ with $p=0.9$, hence increasing average refractory time $n-0.1$ and fixed standard deviation ($\sigma =0.35$). The average was computed over runs and network realizations. Horizontal error bars display the standard deviation $\sigma$ of the refractory time, while vertical error bars are computed from the simulation (dispersion over runs and network realizations).

Figure 16

Validation of the approximate solution Eq. (C3). Comparison of the numerical solution of the stationary mean-field equations, Eq. (C2) (continuous line), for the excitation density, and the approximation $c^{*}\left(E\right)=(1-1/\langle k\rangle )/(1+1/p)$, Eq. (C3) (dotted line), as a function of $p$, for $\langle k\rangle =2.4$. Note the scale on the vertical axis.

Figure 17

Among the features of the cycle length distribution, lifetime correlates best with the average cycle length. Correlation coefficient of the average lifetime with the average cycle length (full black line), with the total number of cycles (dash-dotted blue line), and with the standard deviation of the cycle length distribution (dashed gray line) for different recovery probabilities.

Figure 18

Dependence of the average lifetime on the average cycle length for the modified model. Average lifetime of the network activity for different model realizations (200 different initial conditions for each network, and 200 different network realizations) as a function of the average number of cycles in the graph, for the four models $M_n$, $n=1,2,3,4$.

Figure 19

Trade-off between cycle recoverability and vulnerability to external excitations. The figure displays the average number of complete excitation turns achieved by $L$-cycles, Eq. (1) as a function of their length $L$. The average is performed over simulation runs (200 different initial conditions), then over $L$-cycles, then over network realizations. The networks were grouped in three classes according to their average cycle length $\langle L\rangle$: small, $8\le \langle L\rangle <16$, 40 networks; intermediate, $16\le \langle L\rangle <18$, 35 networks; large, $18\le \langle L\rangle <21$, 25 networks.