Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants

TL;DR

This paper derives exact analytical formulas for the cumulants of neuronal membrane potentials modeled by multivariate Hawkes processes, enabling improved prediction and analysis of neuronal spike train statistics over time.

Contribution

It introduces closed-form expressions for cumulants in a multivariate Hawkes model, enhancing statistical analysis and density estimation of neuronal activity.

Findings

Analytical cumulant formulas outperform Monte Carlo estimates

Provides computational tools for neuronal spike train analysis

Enables sensitivity analysis of neuronal models

Abstract

We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time, and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations, and computer codes based on combinatorial recursions are included.