Macroscopic self-oscillations and aging transition in a network of synaptically coupled quadratic integrate-and-fire neurons

We analyze the dynamics of a large network of coupled quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rate and the mean membrane potential, which are exact in the infinite-size limit. The bifurcation analysis of the reduced equations reveals a rich scenario of asymptotic behavior, the most interesting of which is the macroscopic limit-cycle oscillations. It is shown that the finite width of synaptic pulses is a necessary condition for the existence of such oscillations. The robustness of the oscillations against aging damage, which transforms spiking neurons into nonspiking neurons, is analyzed. The validity of the reduced equations is confirmed by comparing their solutions with the solutions of microscopic equations for the finite-size networks.

Figure 1

Comparison of dynamics of the synaptic variable $S\left(t\right)$ estimated from the microscopic model (5) and (6) (blue solid curve) and the reduced system (20) and (21) (green dashed curve). Parameters are $N=10000,\mathrm{\Delta }=1,\overline{\eta }=0,V_{\mathrm{th}}=50,V_s=75$, and $K=20$.

Figure 2

Two-parameter bifurcation diagram of the macroscopic model (22). The bifurcation in the parameter space $(\overline{\eta },J)$ for the fixed parameter $V_{\mathrm{th}}=50$. The blue solid curve shows the saddle-node bifurcation, the red dashed curve the Andronov-Hopf bifurcation, and the green connected points the homoclinic bifurcation. The dotted magenta curve separates the regions with a stable node and a stable focus. The points BT and CP denote the Bogdanov-Takens and the cusp bifurcations, respectively. The regions marked by Roman numerals correspond to the following set of fixed points and invariant curves: (I) a stable node; (II) a saddle, an unstable focus, and an stable node; (III) an unstable focus and a stable limit cycle; (IV) a stable focus; (V) a stable node, a stable focus, and a saddle; and (VI) a stable node, a saddle, an unstable focus, and a stable limit cycle. Typical phase portraits of the system in the marked regions are depicted in the right-hand side of the figure.

Figure 3

An evolution of the Androv-Hopf bifurcation curve in the $(\overline{\eta },J)$ plane with the increase of the threshold voltage $V_{\mathrm{th}}$.

Figure 4

Asymptotic values of the synaptic variable $S$ multiplied by $V_{\mathrm{th}}$ as a function of the parameter $\overline{\eta }$. The red solid curve represents the stable fixed points, the blue dashed curve shows unstable fixed points, and the green (light) solid curve denote the minimal and maximal values of the periodically oscillating variable $S\left(t\right)V_{\mathrm{th}}$. The curves are derived from the macroscopic model (22), while the symbols show the results obtained from numerical simulations of QIF neurons. Squares denote the averaged stationary values of $S{V}_{\mathrm{th}}$, while the asterisks the averaged minimal and maximal values of the oscillating variable $S\left(t\right)V_{\mathrm{th}}$. Parameters are $N=10000,V_{\mathrm{th}}=50$, (a) $J=25$, (b) $J=20$, and (c) $J=15$.

Figure 5

Dynamics of the firing rate $r$ (top graphics in each panel) and the phases ${\theta }_j$ presented by colors (bottom graphics in each panel) of $N=1000$ synaptically coupled QIF neurons. Parameters are $V_{\mathrm{th}}=50,J=15$, (a) $p=0.01$ $\left(\overline{\eta }=31.82\right)$, (b) $p=0.05$ $\left(\overline{\eta }=6.31\right)$, (c) $p=0.3$ $\left(\overline{\eta }=0.73\right)$, and (d) $p=0.92$ $\left(\overline{\eta }=-3.89\right)$.

Figure 6

(a) The proportion $P$ of inactive neurons in the coupled network as a function of the coupling strength $J$ for different values of the parameter $\overline{\eta }$. Solid curves represent analytical results obtained from Eqs. (33) and (34), while the symbols show the results of numerical simulation of the microscopic model with $N=10000$ neurons. The squares and asterisks for $\overline{\eta }=-5$ correspond to the increase and decrease of the coupling strength, respectively. The vertical dotted lines indicate the onset of macroscopic oscillations. They correspond to the following values of the coupling strength: $J_c^{\left(1\right)}=12.67$ $\left(\overline{\eta }=5\right)$, $J_c^{\left(2\right)}=14.68$ $\left(\overline{\eta }=0\right)$, and $J_c^{\left(3\right)}=17.22$ $\left(\overline{\eta }=-5\right)$. The threshold voltage is $V_{\mathrm{th}}=50$. (b) The proportion $P$ of inactive neurons in the coupled network as a function of the proportion $p$ of inactive neurons in the absence of coupling for different values of the coupling strength $J$. Solid curves show analytical results obtained from Eqs. (33) and (34), while the symbols correspond to numerical simulation of the microscopic model. The vertical dotted lines indicate the critical values $p_c$ at which the oscillations cease to exist: $p_c^{\left(1\right)}=0.006(J=5),p_c^{\left(2\right)}=0.71(J=15)$, and $p_c^{\left(3\right)}=0.94(J=25)$.

Figure 7

Bifurcation diagram in the $\left(J,p\right)$ plane. The colored region corresponds to macroscopic oscillations of the firing rate and the mean membrane potential. The border of this region depicted by the red solid curve defines the dependence of the robustness parameter $p_c$ on the coupling strength $J$. The inset shows an enlarged region of the main figure. The threshold voltage is $V_{\mathrm{th}}=50$.

Figure 8

Bifurcation diagrams in the $\left(V_{\mathrm{th}},p\right)$ plane for different values of the coupling strength $J$. In the colored regions, there are macroscopic oscillations. The borders of the regions depicted by the red solid curves define the dependence of the parameter $p_c$ on the threshold voltage $V_{\mathrm{th}}$.

Figure 9

Two-parameter bifurcation diagram of the macroscopic model (21). The bifurcation in the parameter space $(\overline{\eta },K)$ for the fixed parameters $V_{\mathrm{th}}=50$ and $V_s=75$. All denotations are the same as in Fig. 2.

Figure 10

Bifurcation diagram of the macroscopic model (21) in the $\left(K,p\right)$ plane. The upper axis shows the values of the rescaled coupling strength $J=K{V}_s/V_{\mathrm{th}}$ in order to compare the results with those presented in Fig. 7. The colored region corresponds to macroscopic oscillations of the firing rate and the mean membrane potential. The border of this region depicted by the red solid curve defines the dependence of the robustness parameter $p_c$ on the coupling strength $K$. All parameters are the same as in Fig. 9.