The macroscopic dynamics of large populations of neurons can be mathematically analyzed using low-dimensional firing-rate or neural-mass models. However, these models fail to capture spike synchronization effects of stochastic spiking neurons such as the non-stationary population response to rapidly changing stimuli. Here, we derive low-dimensional firing-rate models for homogeneous populations of general renewal-type neurons, including integrate-and-fire models driven by white noise. Renewal models account for neuronal refractoriness and spike synchronization dynamics. The derivation is based on an eigenmode expansion of the associated refractory density equation, which generalizes previous spectral methods for Fokker-Planck equations to arbitrary renewal models. We find a simple relation between the eigenvalues, which determine the characteristic time scales of the firing rate dynamics, and the Laplace transform of the interspike interval density or the survival function of the renewal process. Analytical expressions for the Laplace transforms are readily available for many renewal models including the leaky integrate-and-fire model. Retaining only the first eigenmode yields already an adequate low-dimensional approximation of the firing-rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset. We explicitly demonstrate the validity of our model for a large homogeneous population of Poisson neurons with absolute refractoriness, and other renewal models that admit an explicit analytical calculation of the eigenvalues. The here presented eigenmode expansion provides a systematic framework for novel firing-rate models in computational neuroscience based on spiking neuron dynamics with refractoriness.
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