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Populations of spiking neuron models have densities of their microscopicvariables (e.g., single-cell membrane potentials) whose evolution fully capturethe collective dynamics of biological networks, even outside equilibrium.Despite its general applicability, the Fokker-Planck equation governing suchevolution is mainly studied within the borders of the linear response theory,although alternative spectral expansion approaches offer some advantages in thestudy of the out-of-equilibrium dynamics. This is mainly due to the difficultyin computing the state-dependent coefficients of the expanded system ofdifferential equations. Here, we address this issue by deriving analyticexpressions for such coefficients by pairing perturbative solutions of theFokker-Planck approach with their counterparts from the spectral expansion. Atight relationship emerges between several of these coefficients and theLaplace transform of the inter-spike interval density (i.e., the distributionof first-passage times). `Coefficients' like the current-to-rate gain function,the eigenvalues of the Fokker-Planck operator and its eigenfunctions at theboundaries are derived without resorting to integral expressions. For the leakyintegrate-and-fire neurons, the coupling terms between stationary andnonstationary modes are also worked out paving the way to accuratelycharacterize the critical points and the relaxation timescales in networks ofinteracting populations.

A `Rosetta stone' for the population dynamics of spiking neuron networks