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

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.

Figures (2)

• Figure 1: Schematic representation of a single two-phase QIF \ref{['eq:micro']} voltage time series in panel $(a)$ and the ansatz distribution \ref{['eq:distribution']} in panel $(b)$. Contributions of the first phase are marked with green and of the second phase with orange. In panel $(b)$ the full distribution $\rho(v)$ is depicted with black.
• Figure 2: Example simulation comparing the dynamics from a large ($N=10^6$), finite ensemble of microscopic equations \ref{['eq:micro_ex']} (blue) and exact macroscopic equation \ref{['eq:macro_ex']} (red). In panel $(a)$ the distribution of voltages \ref{['eq:distribution']}, in panel $(b)$ voltage traces for some values of the heterogeneity $\eta_j$, and in panels $(c)$ and $(d)$ the mean voltage $V$ and firing rate $R$ respectively in time. Parameters are: $v_\text{min}=-3$, $v_\text{max}=13$, $J=3$, $g=0.05$, $I=-0.2$, $\eta_0=0$, and $\Delta=0.05$.