Distinct dynamical behavior in Erdős-Rényi networks, regular random networks, ring lattices, and all-to-all neuronal networks

Neuronal network dynamics depends on network structure. In this paper we study how network topology underpins the emergence of different dynamical behaviors in neuronal networks. In particular, we consider neuronal network dynamics on Erdős-Rényi (ER) networks, regular random (RR) networks, ring lattices, and all-to-all networks. We solve analytically a neuronal network model with stochastic binary-state neurons in all the network topologies, except ring lattices. Given that apart from network structure, all four models are equivalent, this allows us to understand the role of network structure in neuronal network dynamics. While ER and RR networks are characterized by similar phase diagrams, we find strikingly different phase diagrams in the all-to-all network. Neuronal network dynamics is not only different within certain parameter ranges, but it also undergoes different bifurcations (with a richer repertoire of bifurcations in ER and RR compared to all-to-all networks). This suggests that local heterogeneity in the ratio between excitation and inhibition plays a crucial role on emergent dynamics. Furthermore, we also observe one subtle discrepancy between ER and RR networks, namely, ER networks undergo a neuronal activity jump at lower noise levels compared to RR networks, presumably due to the degree heterogeneity in ER networks that is absent in RR networks. Finally, a comparison between network oscillations in RR networks and ring lattices shows the importance of small-world properties in sustaining stable network oscillations.

Figure 1

Steady-state neuronal activity $\rho$ as a function of the level of noise $\langle n\rangle /c$ and $\langle \eta \rangle$ in ER (left column), RR (middle column), and all-to-all networks (right column). These steady states are the result of the numerical integration of Eqs. (14, 15, 16). Each row corresponds to networks with different fractions of excitatory neurons: (a)–(c) $g_e=0.74$, (d)–(f) $g_e=0.75$, and (g)–(i) $g_e=0.76$. The dashed lines represent upper metastable states in bistability regions where $\rho$ may take low or high activity values depending on the initial conditions.

Figure 2

Noise–$\alpha$ planes of the phase diagram of the neuronal network models. Left, middle, and right columns correspond, respectively, to ER, RR, and all-to-all networks. Each row represents networks with different fractions of excitatory neurons: (a)–(c) $g_e=0.74$, (d)–(f) $g_e=0.75$, and (g)–(i) $g_e=0.76$. There are four regions of activity: (I) low neuronal activity; (II) bistability region; (III) neuronal network oscillations; and (IV) high neuronal activity with (a) exponential relaxation and (b) damped oscillations. All-to-all networks have a region I+IVa which contains a continuum from low to high activity as a function of increasing noise intensity $\eta$. The black and yellow solid lines are the numerical solutions of Eq. (21), whereas the black dashed lines are the numerical solutions of Eq. (22). The black solid lines correspond to supercritical Hopf bifurcations and the yellow solid lines represent subcritical Hopf bifurcations. The red lines correspond to saddle-node bifurcations determined by Eq. (23).

Figure 3

Excitatory activity ${\rho }_e$ as a function of time in ER (left column), and all-to-all networks (right column). Panels (a) and (b) display low activity from region I in Fig. 2, $(\langle n\rangle /c,\alpha )=(\langle \eta \rangle ,\alpha )=(0.015,0.7)$; panels (c) and (d) represent high activity from region IVb in ER networks and IVa in all-to-all networks, $(\langle n\rangle /c,\alpha )=(\langle \eta \rangle ,\alpha )=(0.05,0.9)$; and panels (e) and (f) show network oscillations from region III, $(\langle n\rangle /c,\alpha )=(\langle \eta \rangle ,\alpha )=(0.03,0.7)$. The black lines are the numerical solution of Eq. (1) for each network topology, and the blue triangles represent numerical simulations of the model (number of neurons $N={10}^5$). We used a fraction of excitatory neurons $g_e=0.75$.

Figure 4

Network oscillations in RR networks (left column), and in ring lattices (right column). As in Fig. 3, the blue triangles represent numerical simulations of the model in finite networks. The black lines are the numerical solution of Eq. (1) for RR networks. The same numerical solutions are plotted as dashed lines in the right column for comparison with the simulations in finite ring lattices. Panels (a) and (b) correspond to networks with size $N={10}^4$, whereas panels (c) and (d) display oscillations in networks with size $N={10}^5$. We used the same parameters in all the panels: $(\langle n\rangle /c,\alpha )=(0.03,0.7)$, and a fraction of excitatory neurons $g_e=0.75$.