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

TL;DR

This paper develops an analytical framework linking spectral expansion and Fokker-Planck methods to better understand the out-of-equilibrium dynamics of spiking neuron networks, enabling precise characterization of their collective behavior.

Contribution

It derives explicit formulas for coefficients in spectral expansions of the Fokker-Planck equation, improving analysis of neuron population dynamics beyond linear response.

Findings

Analytic expressions for spectral coefficients derived without integral formulas

Relationship established between coefficients and inter-spike interval distribution

Enhanced characterization of critical points and relaxation times in neuron networks

Abstract

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equation governing such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the inter-spike interval density (i.e., the distribution of first-passage times). `Coefficients' like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.