Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models—also known as firing rate models or firing rate equations—to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.

Figure 1

Strong electrical coupling suppresses oscillatory activity for negative mean currents $\overline{\eta }<0$. The panels show the time series of the membrane voltages of $N=2$ electrically coupled QIF neurons. (a) Self-oscillatory neuron with ${\eta }_1={\pi }^2$; (b) quiescent neuron with ${\eta }_2=-2{\pi }^2$. The two neurons are uncoupled (black thin curves, $g=0$), weakly coupled (red curves, $g=1$), or strongly coupled (blue thick curves, $g=6$). We used $\tau =10\mathrm{ms}$ and $J=0$.

Figure 2

The sign of the mean current $\overline{\eta }$ determines the behavior of the fixed points of the FREs (5) with electrical coupling only, $J=0$. The panels show the nullclines of the FREs (5) with only electrical coupling ($J=0$) for negative ($\overline{\eta }/\mathrm{\Delta }=-2$) and positive ($\overline{\eta }/\mathrm{\Delta }=0.5$) values of $\overline{\eta }$ [panels (a) and (b), respectively], and $g/\sqrt{\mathrm{\Delta }}=0,2,4$. The black points correspond to the intersections of $r$ nullclines ($\dot{r}=0$, red) and $v$ nullclines ($\dot{v}=0$, gray) and are fixed points of Eqs. (5).

Figure 3

The bifurcation diagrams of the FREs (5) for networks with (left) electrical and (right) chemical coupling are qualitatively different. The top panels show the scaled firing rate $r_{*}/\left(\pi \tau \sqrt{\mathrm{\Delta }}\right)$ vs the ratio of (a) electrical $g/\sqrt{\mathrm{\Delta }}$ and (b) chemical $J/\left(\pi \sqrt{\mathrm{\Delta }}\right)$ synaptic strengths. Panels (c) and (d) show the same bifurcation diagrams for the scaled mean membrane potential $v_{*}/\sqrt{\mathrm{\Delta }}$.

Figure 4

Electrical coupling promotes collective synchrony. The inclusion of inhibitory coupling degrades synchrony and slows down oscillations. The figure shows the time series of the mean firing rate $r$ of a network of $N={10}^4$ QIF neurons with dynamics (1) (black) and of the FREs (5) (red). Panel (a) shows collective oscillations (frequency $f\approx 30.1\mathrm{Hz}$) of a network with gap junctions only ($J=0$). Panel (b) corresponds to a network with both gap junctions and inhibitory chemical coupling ($J=-\pi$), which both slows down ($f\approx 23.6\mathrm{Hz}$) and reduces the amplitude of collective oscillations. Parameters: $g=3,\overline{\eta }=1,\tau =10\mathrm{ms}$, and $\mathrm{\Delta }=1$.

Figure 5

In the absence of chemical coupling (black), electrical coupling $g$ has little effect on the frequency of the oscillations. Excitatory (red) and inhibitory (blue) couplings speed up and slow down collective oscillations. These effects tend to disappear for strong electrical coupling. The figure shows the frequency of the oscillations $f$ as a function of the strength of electrical coupling $g$ in networks with excitation $J=\pi$, inhibition $J=-\pi$, and without chemical coupling $J=0$. The symbols ($\circ$) are frequencies obtained from numerical simulations of a network of $N={10}^4$ QIF neurons Eqs. (1). The solid lines are numerically obtained frequencies from the FREs (5). The dotted lines correspond to the Hopf frequency given by Eq. (8). Parameters: $\overline{\eta }=1,\tau =10\mathrm{ms}$, and $\mathrm{\Delta }=1$.

Figure 6

The phase diagram of the FREs (5) for electrical coupling only ($J=0$) is characterized by the presence of three codimension-2 bifurcation points [TB; saddle-node separatix loop (SNSL), and cusp], all located at $\overline{\eta }\ge 0$. The region of synchronization (Sync) is limited by supercritical Hopf (red), SNIC (black), and homoclinic (hom) (green) bifurcations. The dashed line: Focus-node (FN) boundary of the asynchronous state. Panel (b) Enlargement of the region near the three codimension-2 points. There are two small regions of bistability among asynchronous, LAS, and asynchronous persistent states and between LAS and Sync. Two SN bifurcations are created at a cusp point at $[1/\left(3\sqrt{3}\right),4\sqrt{2}/3^{3/4}]\approx (0.192,2.482)$. The upper SN line meets the hom bifurcation in a SNSL point. At this point, the upper SN becomes a SNIC bifurcation. The other SN bifurcation tangentially meets the homoclinic and the Hopf lines at a TB point at $(0,2\sqrt{2})\approx (0,2.828)$. The Hopf boundary corresponds to Eq. (7). SN and SNIC and focus-node boundaries are obtained in parametric and explicit forms, respectively, in Appendix pp3. The homoclinic boundary has been obtained numerically. The symbol $\times$ indicates the parameter value considered in Fig. 4.

Figure 7

The phase diagram for networks with only chemical coupling panel (a) is characterized by the presence of a cusp bifurcation point. The inclusion of electrical coupling, panels (b)–(d), transforms the cusp bifurcation scenario into that of Fig. 6, characterized by the presence of three codimension-2 bifurcation points. The panels show the phase diagrams of the FREs (5) with chemical coupling ($J>0$: excitatory; $J<0$: inhibitory) for (a) $g/\sqrt{\mathrm{\Delta }}=0$, (b) $g/\sqrt{\mathrm{\Delta }}=1$, (c) $g/\sqrt{\mathrm{\Delta }}=3$, and (d) $g/\sqrt{\mathrm{\Delta }}=5$. The Hopf boundaries (red lines) are straight lines given by Eq. (7). SN and SNIC (black lines) and focus-node (dashed) boundaries are obtained in parametric form in Appendix pp3. Hopf and SN boundaries meet at a TB bifurcation point. To lighten the diagrams, cusp, SNSL, and homoclinic bifurcations are not shown. Symbols $\times$ and $+$ indicate the parameter values considered in Fig. 4.