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Chemical and electrical synapses shape the dynamics of neuronal networks.Numerous theoretical studies have investigated how each of these types ofsynapses contributes to the generation of neuronal oscillations, but theircombined effect is less understood. This limitation is further magnified by theimpossibility of traditional neuronal mean-field models ---also known as firingrate models, or firing rate equations--- to account for electrical synapses.Here we introduce a novel firing rate model that exactly describes the meanfield dynamics of heterogeneous populations of quadratic integrate-and-fire(QIF) neurons with both chemical and electrical synapses. The mathematicalanalysis of the firing rate model reveals a well-established bifurcationscenario for networks with chemical synapses, characterized by a codimension-2Cusp point and persistent states for strong recurrent excitatory coupling. Theinclusion of electrical coupling generally implies neuronal synchrony by virtueof a supercritical Hopf bifurcation. This transforms the Cusp scenario into abifurcation scenario characterized by three codimension-2 points (Cusp,Takens-Bogdanov, and Saddle-Node Separatrix Loop), which greatly reduces thepossibility for persistent states. This is generic for heterogeneous QIFnetworks with both chemical and electrical coupling. Our results agree withseveral numerical studies on the dynamics of large networks of heterogeneousspiking neurons with electrical and chemical coupling.

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