A reduction methodology for fluctuation driven population dynamics

TL;DR

This paper introduces a pseudo-cumulants expansion to overcome divergence issues in Lorentzian-based models, enabling a reduction methodology for heterogeneous neural networks with noise, extending previous mean-field approaches.

Contribution

The paper presents a novel pseudo-cumulants expansion that generalizes mean-field reduction methods for noisy, heterogeneous neural populations.

Findings

Enables analytic treatment of Lorentzian distributions with deformations.

Provides a reduction methodology for complex neural network dynamics.

Extends mean-field models to include extrinsic and endogenous noise.

Abstract

Lorentzian distributions have been largely employed in statistical mechanics to obtain exact results for heterogeneous systems. Analytic continuation of these results is impossible even for slightly deformed Lorentzian distributions, due to the divergence of all the moments (cumulants). We have solved this problem by introducing a `pseudo-cumulants' expansion. This allows us to develop a reduction methodology for heterogeneous spiking neural networks subject to extrinsinc and endogenous noise sources, thus generalizing the mean-field formulation introduced in [E. Montbri\'o et al., Phys. Rev. X 5, 021028 (2015)].