Effects of degree distributions in random networks of type-I neurons

We consider large networks of theta neurons and use the Ott-Antonsen ansatz to derive degree-based mean-field equations governing the expected dynamics of the networks. Assuming random connectivity, we investigate the effects of varying the widths of the in- and out-degree distributions on the dynamics of excitatory or inhibitory synaptically coupled networks and gap junction coupled networks. For synaptically coupled networks, the dynamics are independent of the out-degree distribution. Broadening the in-degree distribution destroys oscillations in inhibitory networks and decreases the range of bistability in excitatory networks. For gap junction coupled neurons, broadening the degree distribution varies the values of parameters at which there is an onset of collective oscillations. Many of the results are shown to also occur in networks of more realistic neurons.

Figure 1

(a) $s$ for the reduced model (17) and (18) (blue curve) and for the full model (1, 2, 3) (dots) for $p_{\text{out}}$ being uniform on [10,190], [50,150], and [90,110] (different colors). $p_{\text{in}}$ is uniform on [95,105] (i.e., $\sigma =5$). A different realization of the ${\eta }_i$ was used for the different networks. (b) As for (a), but now $p_{\text{in}}$ is uniform on [50,150] ($\sigma =50$). For (1, 2, 3), $N=500$ neurons were used and the initial conditions were ${\theta }_i=u_i=0$, and for (17) and (18) we used $b=1$ and $s=0$.

Figure 2

Hopf bifurcation curve for a fixed point of (17) and (18). A stable periodic orbit exists to the left of the curve, and a stable fixed point exists to the right. Other parameters are ${\eta }_0=1,\mathrm{\Delta }=0.05,K=-2,\langle k\rangle =100$. $p_{\text{in}}$ is uniform on $[100-\sigma ,100+\sigma ]$.

Figure 3

A beta distribution of in-degrees. A Hopf bifurcation curve for a fixed point of (17) and (18). A stable periodic orbit exists to the left of the curve, and a stable fixed point exists to the right. Other parameters are ${\eta }_0=1,\mathrm{\Delta }=0.05,K=-2,\langle k\rangle =100$. The inset shows the beta distribution on [50,150] with $\alpha =3,20$.

Figure 4

Results for a Morris-Lecar model. $\tau$ is the synaptic timescale, and the in-degree distribution is uniform on $[100-\sigma ,100+\sigma ]$. Color shows the standard deviation of $\widehat{s}$ over 5 s of simulated time, having discarded the first 45 s. The same ${\eta }_i$ were used for each simulation. We used $N=500$ neurons.

Figure 5

$s$ at fixed points of (17) and (18) for $\sigma =10$ (red) and $\sigma =90$ (blue). Solid curves are stable, dashed unstable. Parameters: $\mathrm{\Delta }=0.05,\tau =1,K=5,\langle k\rangle =100$, uniform in-degree on $[100-\sigma ,100+\sigma ]$.

Figure 6

Curves of saddle-node bifurcations of fixed points of (17) and (18). The network is bistable between the curves and has a single attractor outside this region. Figure 5 corresponds to horizontal “slices” through this figure at $\sigma =10$ and 90. Parameters: $\mathrm{\Delta }=0.05,\tau =1,K=5,\langle k\rangle =100$, uniform in-degree on $[100-\sigma ,100+\sigma ]$.

Figure 7

Approximate steady states for a Morris-Lecar model. For each of the two networks, $I_0$ was quasistatically increased up and then down. Both networks show bistability for a range of $I_0$ values, and vertical jumps at apparent saddle-node bifurcations. Compare with Fig. 5. Parameters: $\langle k\rangle =100$, uniform in-degree on $[100-\sigma ,100+\sigma ]$, out-degree is uniform on [50,150]. The same ${\eta }_i$ were used for each simulation.

Figure 8

SNIC bifurcation curve of a fixed point of (41) and (42). A stable fixed point exists to the left of the curve, and stable periodic oscillations exist to the right. Parameters: $\mathrm{\Delta }=0.01,g=0.4,\langle k\rangle =100$, $P\left(k\right)$ is a uniform distribution on $[100-\sigma ,100+\sigma ]$.

Figure 9

Hopf bifurcation curve of a fixed point of (41) and (42). A stable fixed point exists to the left, and stable periodic oscillations exist to the right. Parameters: $\mathrm{\Delta }=0.05,\langle k\rangle =100,{\eta }_0=0.2$, $P\left(k\right)$ is uniform distribution on $[100-\sigma ,100+\sigma ]$.

Figure 10

Evidence of a supercritical Hopf bifurcation in a network of gap-junction coupled Morris-Lecar neurons. On the vertical axis we plot the standard deviation of $\widehat{s}$ (the mean of the $s_i$) over a period of 10 s, having discarded the first 10 s as transients.

Figure 11

SNIC bifurcations for gap-junction coupled Morris-Lecar neurons. (a) $I_i$ chosen from a uniform distribution on $[-1/2,1/2]$ (blue circles, solid curve) or a normal distribution with center zero and standard deviation $1/3$ (red triangle, dashed curve). (b) $I_i$ chosen from a uniform distribution on $[-1/8,1/8]$ (blue circles, solid curve) or a normal distribution with center zero and standard deviation $1/10$ (red triangle, dashed curve). The curves result from fitting the values of $I_0$ as a cubic function of $\sigma$ and are a guide to the eye. Parameters: $N=2500,\epsilon =0.3$.

Figure 12

Evidence of Hopf bifurcations in networks of gap-junction coupled Morris-Lecar neurons. (a) $I_i$ taken from a unit Gaussian distribution. The Hopf bifurcations appear to be supercritical. (b) $I_i$ taken from a uniform distribution on $[-1,1]$. The Hopf bifurcation for $\sigma =10$ appears to be subcritical, as explained in the text. For each panel, on the vertical axis we plot the standard deviation of $\widehat{s}$ (the mean of $s_i$) over a period of 10 s, having discarded the first 10 s as transients. $N=2500$.