Multiplicatively interacting point processes and applications to neural modeling

TL;DR

This paper introduces a nonlinear extension of the Hawkes process enabling inhibitory interactions, providing a new mathematical model for neural networks that captures stability and robust winner-takes-all dynamics, and relates to generalized linear models.

Contribution

A novel nonlinear Hawkes process model allowing inhibitory couplings, with analytical stability analysis and applications to neural network modeling.

Findings

The model accurately approximates neural activity rates.

The stability analysis identifies conditions for robust network behavior.

The winner-takes-all network demonstrates wide parameter robustness.

Abstract

We introduce a nonlinear modification of the classical Hawkes process, which allows inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for recurrent networks of spiking neurons with exponential transfer functions. The expected rates of all neurons in the network are approximated by a first-order differential system. We study the stability of the solutions of this equation, and use the new formalism to implement a winner-takes-all network that operates robustly for a wide range of parameters. Finally, we discuss relations with the generalised linear model that is widely used for the analysis of spike trains.