Low-dimensional model for adaptive networks of spiking neurons

We investigate a large ensemble of quadratic integrate-and-fire neurons with heterogeneous input currents and adaptation variables. Our analysis reveals that, for a specific class of adaptation, termed quadratic spike-frequency adaptation, the high-dimensional system can be exactly reduced to a low-dimensional system of ordinary differential equations, which describes the dynamics of three mean-field variables: the population's firing rate, the mean membrane potential, and a mean adaptation variable. The resulting low-dimensional firing rate equations (FREs) uncover a key generic feature of heterogeneous networks with spike-frequency adaptation: Both the center and width of the distribution of the neurons' firing frequencies are reduced, and this largely promotes the emergence of collective synchronization in the network. Our findings are further supported by the bifurcation analysis of the FREs, which accurately captures the collective dynamics of the spiking neuron network, including phenomena such as collective oscillations, bursting, and macroscopic chaos.