Two-phase quadratic integrate-and-fire neurons: Exact low-dimensional description for ensembles of finite-voltage neurons

TL;DR

This paper introduces a two-phase quadratic integrate-and-fire neuron model that avoids unphysical voltage divergence while maintaining an exact low-dimensional description, enabling realistic and analytically tractable collective neural dynamics.

Contribution

The authors propose a novel two-phase QIF neuron model that is biologically plausible, exactly solvable, and compatible with existing mean-field frameworks, improving upon the standard QIF model.

Findings

Removes unphysical voltage divergence in QIF neurons

Maintains an exact low-dimensional description in the thermodynamic limit

Provides compact, analytical expressions for collective neural quantities

Abstract

We introduce a two-phase quadratic integrate-and-fire (QIF) neuron whose membrane potential evolves according to two alternating Riccati equations within finite bounds. This simple extension removes the unphysical voltage divergence of the standard QIF model while producing realistic spike waveforms. Despite this modification, the system retains an exact low-dimensional description in the thermodynamic limit, governed by a single complex Riccati equation. Expressions for collective quantities such as the firing rate and mean voltage remain compact and analytically tractable. Because the formalism preserves the mathematical structure of the standard QIF ensemble, it inherits its many generalizations and can serve as a drop-in replacement in existing mean-field frameworks, providing a more biologically plausible yet still exactly solvable neuronal model.