Effects of distance-dependent delay on small-world neuronal networks

We study firing behaviors and the transitions among them in small-world noisy neuronal networks with electrical synapses and information transmission delay. Each neuron is modeled by a two-dimensional Rulkov map neuron. The distance between neurons, which is a main source of the time delay, is taken into consideration. Through spatiotemporal patterns and interspike intervals as well as the interburst intervals, the collective behaviors are revealed. It is found that the networks switch from resting state into intermittent firing state under Gaussian noise excitation. Initially, noise-induced firing behaviors are disturbed by small time delays. Periodic firing behaviors with irregular zigzag patterns emerge with an increase of the delay and become progressively regular after a critical value is exceeded. More interestingly, in accordance with regular patterns, the spiking frequency doubles compared with the former stage for the spiking neuronal network. A growth of frequency persists for a larger delay and a transition to antiphase synchronization is observed. Furthermore, it is proved that these transitions are generic also for the bursting neuronal network and the FitzHugh-Nagumo neuronal network. We show these transitions due to the increase of time delay are robust to the noise strength, coupling strength, network size, and rewiring probability.

Figure 1

Typical connection of a small-world network and the explanation for the time delay related to distance. (a) A ring structured small-world network is displayed with vertices $N=20$ and connections to $k=4$ nearest neighbors, and the rewiring probability $p=0.2$. (b) The explanation of the time delay related to distance between neurons in the network. From the figure, the distance between neurons $i$ and $k$ is larger than that of neurons $i$ and $j$, as a consequence, the time delay from neuron $i$ to $k$ is larger than the latter.

Figure 2

Spatiotemporal pattern plots of $x^i\left(n\right)$ obtained for different time delays. From (a) to (e), the element time delay ${\tau }_e$ equals 0, 700, 1500, 3400, and 4500 (for convenience, we set $r=1$). (f) is the enlargement of (d), which shows the regular zigzag pattern. The color is linearly divided into ten levels with white depicting 0.0 and black depicting $–1.5$ values. Other parameters are $N=150, w=0.01, p=0.1$, and $D=0.02$.

Figure 3

Time series for neurons 40 and 41 and the corresponding similarity function [parameters are the same as in Fig. 2]. (a) The time series for neurons 40 and 41 are characterized by solid blue line and dashed red line, respectively. They are qualitatively the same except for a phase lag. (b) From the similarity function, the minimum is obtained at a time shift of 75, where $S\left(\tau \right)$ is less than 0.05, that shows a lag synchronization between the two neurons.

Figure 4

The interspike intervals (ISIs) of $x^i\left(n\right)$ on different element time delays. The ISIs for different delays were obtained by averaging over time ($1\times {10}^4$ iteration steps) and space (all neurons) (computations for ISIs and IBIs below are done in the same way). There were roughly four stages to be seen within the delay range in the figure: 0–1100, 1100–2100, 2200–3500, and 4100–4800. The first stage: delay intermittently disturbed the firing behaviors; the second stage: delay-induced irregular zigzag patterns; the third stage: delay-induced regular zigzag patterns; finally, delay-induced antiphase synchronizations. Parameters are the same as in Fig. 2.

Figure 5

Spatiotemporal patterns and ISIs of $x^i\left(n\right)$ for the random delayed neuronal network with different time delays. From (a) to (e), the common time delay ${\tau }_0$ equals 0, 50, 130, 230, and 450. (f) is the ISI calculated for different delays. The color is linearly divided into ten levels with white depicting 0.0 and black depicting $–1.5$. Other parameters are the same as in Fig. 2.

Figure 6

ISIs of $x^i\left(n\right)$ for different distance adjusting parameter $t_d$ with different element time delays. (a) ISI for $t_d=0$ and $t_d=1$. Inset: left: spatiotemporal patterns at ${\tau }_e=2400$; right: spatiotemporal patterns at ${\tau }_e=4100$. (b) ISI for $t_d=0$, 0.1, 0.2, and 0.3. Inset: spatiotemporal patterns at ${\tau }_e=3000$.

Figure 7

Time series, spatiotemporal patterns and interburst intervals (IBIs) of $x^i\left(n\right)$. (a) Typical bursting without time delay. From (b) to (e), spatiotemporal patterns with different element time delays; ${\tau }_e$ equals 0, 150, 1950, and 3300 ($r=4$). (f) is the enlargement of (d), which shows the regular zigzag pattern. The color is linearly divided into ten levels with white depicting 0.5 and black depicting $–2.5$. (g) is the IBI on different element time delays. Parameters: $a=2.9$; others are the same as in Fig. 2.

Figure 8

Spatiotemporal patterns and ISIs of $v^i\left(t\right)$ for FHN neurons. From (a) to (d), spatiotemporal patterns (time ranges from 50 to 60 s and integral step is 0.001) with different element time delays, ${\tau }_e$ equals 0, 1.6, 4, and 7.2 ($r=1$). (c) Exhibits the regular zigzag pattern. The color is linearly divided into ten levels with white depicting 1 and black depicting 0. (e) is the ISI on different element time delays. Other parameters are in the text.

Figure 9

The robustness of other parameters on these firing behaviors and transitions. From (a) to (d), the ISIs of $x^i\left(n\right)$ for network size, rewiring probability, noise strength, and coupling strength with different element time delays. Parameters are the same as in Fig. 2 except the parameter considered.