(A) When a single neuron receives more excitatory input (red) than inhibitory input (blue), its membrane voltage (black line) spikes regularly. In this "mean-driven" model the membrane voltage spends much of its time at or near the spiking threshold (see histogram on right), and spiking is driven by the difference between the means of the excitatory and inhibitory inputs. However, most neurons do not spike as rapidly or regularly as predicted by the "mean-driven" model. (B) The theory of balanced networks proposes that the inhibitory and excitatory inputs to a neuron have approximately the same mean and the same variance. Consequently, spiking is driven by random fluctuations in the inputs, when the excitatory input briefly exceeds the inhibitory input. A signature of this “fluctuation-driven” regime is that the distribution of the membrane voltage is approximately Gaussian (histogram). Fluctuation-driven neurons spike slowly and irregularly, similar to the majority of the neurons in the cortex. (C) The relationship between the input to a neuron (blue curve below the x-axis) and its spiking or firing rate (blue curve to the left of the y-axis) is described by a firing rate versus input (f–i) curve. In 2011 Roxin et al. predicted that an expansive f–i curve (see main text) would convert a Gaussian input into a long-tailed distribution of firing rates in the fluctuation-driven regime. Moreover, according to this theory, if each neuron has a Gaussian input, then the overall population of neurons will also have a Gaussian input, and if each neuron has an expansive f–i curve in the fluctuation-driven regime, then the output of the population will be long-tailed.