Are the input parameters of integrate-and-fire neurons uniquely determined by rate and CV?

Integrate-and-fire (IF) neurons have found widespread applications in computational neuroscience. Particularly important are stochastic versions of these models where the driving consists of a mean input (base current μ) and a fluctuating current (white Gaussian noise of intensity D). Different IF models have been proposed, the firing statistics of which depends nontrivially on the input parameters μ and D. Comparison of these models among each other or with real neurons should be performed at parameters that yield similar basic firing statistics as, for instance, the firing rate and the coefficient of variation (CV) of the interspike interval. However, it is not clear a priori whether for a given firing rate and CV, there is only one unique choice of input parameters for the respective model. Here we review the dependence of rate and CV on input parameters for the perfect, leaky, and quadratic IF neuron models and show analytically that indeed in all three models the firing rate and the CV of the interspike interval distribution uniquely determine the input parameters. For the leaky and quadratic IF models, we use properties of the contour lines for fixed rate and CV and also give simple numerical algorithms leading from given rate and CV to the actual input parameters.