At functional scales, cortical behavior results from the complex interplay of
a large number of excitable cells operating in noisy environments. Such systems
resist to mathematical analysis, and computational neurosciences have largely
relied on heuristic partial (and partially justified) macroscopic models, which
successfully reproduced a number of relevant phenomena. The relationship
between these macroscopic models and the spiking noisy dynamics of the
underlying cells has since then been a great endeavor. Based on recent
mean-field reductions for such spiking neurons, we present here {a principled
reduction of large biologically plausible neuronal networks to firing-rate
models, providing a rigorous} relationship between the macroscopic activity of
populations of spiking neurons and popular macroscopic models, under a few
assumptions (mainly linearity of the synapses). {The reduced model we derive
consists of simple, low-dimensional ordinary differential equations with}
parameters and {nonlinearities derived from} the underlying properties of the
cells, and in particular the noise level. {These simple reduced models are
shown to reproduce accurately the dynamics of large networks in numerical
simulations}. Appropriate parameters and functions are made available {online}
for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley
models
