# Mean-field dynamics of a population of stochastic map neurons

## TL;DR

This paper develops a mean-field model for a population of stochastic map neurons, accurately capturing diverse collective behaviors and responses to stimuli, with strong qualitative and quantitative agreement with the exact system.

## Contribution

It introduces a cumulant-based mean-field approach that effectively models complex neuronal dynamics and responses, extending understanding of stochastic neural populations.

## Key findings

- Model captures stability and bifurcations of the exact system.
- Qualitative and quantitative agreement with the original dynamics.
- Reproduces phase response curves and stimulus responses accurately.

## Abstract

We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis. It is demonstrated that the mean-field model can qualitatively account for stability and bifurcations of the exact system, capturing all the generic forms of collective behavior, including macroscopic excitability, subthreshold oscillations, periodic or chaotic spiking and chaotic bursting dynamics. Apart from qualitative analogies, we find a substantial quantitative agreement between the exact and the approximate system, as reflected in matching of the parameter domains admitting the different dynamical regimes, as well as the characteristic properties of the associated time series. The effective model is further shown to reproduce with sufficient accuracy the phase response curves of the exact system and the assembly's response to external stimulation of finite amplitude and duration.