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

Chemical and electrical synapses shape the collective dynamics of neuronal networks. Numerous theoretical studies have investigated how, separately, each of these type 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 ---often referred to as firing rate models--- to account for electrical synapses. Here we perform a comparative analysis of the dynamics of heterogeneous populations of quadratic integrate-and-fire neurons with chemical, electrical, and both chemical and electrical coupling.

In the thermodynamic limit, we show that the population's mean-field dynamics is exactly described by a system of two ordinary differential equations for the center and the width of the distribution of membrane potentials -or, equivalently, for the population-mean membrane potential and firing rate. These firing rate equations describe, in a unified framework, the collective dynamics of the ensemble of spiking neurons, and reveal that both chemical and electrical coupling are mediated by the population firing rate. Furthermore, while chemical coupling shifts the center of the distribution of membrane potentials, electrical coupling tends to reduce the width of this distribution promoting the emergence of synchronization. The analysis of the firing rate equations allows us to obtain exact formulas for all Saddle-Node and Hopf boundaries, and to construct phase diagrams characterizing the dynamics of the original network of spiking neuron. In networks with instantaneous chemical synapses the phase diagram is characterized by a codimension-two Cusp point, and by the presence of persistent states for strong excitatory coupling. In contrast, the phase diagram for electrically coupled networks is determined by a Takens-Bogdanov codimension-two point, which entails the presence of oscillations and greatly reduces the possibility of persistent states. In this case oscillations arise either via a Saddle-Node-Invariant-Circle bifurcation, or through a supercritical Hopf bifurcation -as shown using weakly nonlinear stability analysis. Finally, we show that the Takens-Bogdanov bifurcation scenario is generically present in networks with both chemical and electrical coupling.