Firing rate of the noisy quadratic integrate-and-fire neuron

We calculate the firing rate of the quadratic integrate-and-fire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, tau(s), to the neuronal time constant, tau(m). We calculate the firing rate exactly in two limits: when the ratio, goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O(tau(s)/tau(m)), which is qualitatively different from that of the leaky integrate-and-fire neuron, where the correction is O(roottau(s)/tau(m)). The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O(tau(m)/tau(s)). By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of tau(s)/tau(m) in the suprathreshold regime-that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when tau(s) becomes large compared to tau(m).