Low-dimensional firing rate dynamics of spiking neuron networks

Starting from a spectral expansion of the Fokker-Plank equation for the membrane potential density in a network of spiking neurons, a low-dimensional dynamics of the collective firing rate is derived. As a result a $n$-order ordinary differential equation for the network activity can be worked out by taking into account the slowest $n$ modes of the expansion. The resulting low-dimensional dynamics naturally takes into account the strength of the synaptic couplings under the hypothesis of a not too fast changing membrane potential density. By considering only the two slowest modes, the firing rate dynamics is equivalent to the one of a damped oscillator in which the angular frequency and the relaxation time are state-dependent. The presented results apply to a wide class of networks of one-compartment neuron models.