Phase Plane Analysis of Firing Patterns in the Adaptive Exponential Integrate-and-Fire Model

TL;DR

The paper addresses how the Adaptive Exponential Integrate-and-Fire ($AdEx$) model reproduces a diverse repertoire of neuronal firing patterns under parametric variation. It applies phase-plane analysis of the two-dimensional $(V,w)$ system, examining nullcline geometry and trajectory segments to link dynamics with spike timing. Six representative patterns—Tonic Spiking, Adapting, Initial Bursting, Bursting, Transient Spiking, and Delayed Spiking—emerge from parameter-driven changes in adaptation and spike-triggered increments, governed by the $V$-nullcline and slow dynamics near nullclines. The work provides a theoretical framework for interpreting firing in simplified neuronal models and informs neuromorphic computing and brain–computer interface applications, guiding future parameter sweeps and experimental validation.

Abstract

The Adaptive Exponential Integrate-and-Fire (AdEx) model is a simplified framework that effectively characterizes neuronal electrical activity. The aim of this paper is to employ phase plane analysis to systematically investigate diverse firing patterns generated by the AdEx model under varying parametric conditions. We first introduce the fundamental equations and parameter configurations of the AdEx model to numerically simulate the six representative firing patterns in the AdEx model. And then we use phase plane analysis to explore the dynamic mechanism of these firing patterns under different input currents and parametric conditions. Our findings demonstrate that the AdEx model can simulate multiple firing patterns, including Tonic Spiking, Adapting, Initial Bursting, Busting, Transient Spiking and Delayed Spiking firing patterns. These results not only advance the understanding of complex electrophysiological phenomena in neurons but also provide theoretical foundations for applications in many fields like neuromorphic computing and brain-computer interfaces.

Figures (12)

• Figure 1: Simulation results of the Tonic Spiking pattern in the AdEx model. The upper figure shows the variation of the membrane potential $V$ over time in the Tonic Spiking pattern, while the lower figure shows the variation of the adaptation current $\omega$ over time in the Tonic Spiking pattern.
• Figure 2: Simulation results of the Adapting pattern in the AdEx model. The upper figure shows the variation of the membrane potential $V$ over time in the Adapting pattern, while the lower figure shows the variation of the adaptation current $\omega$ over time in the Adapting pattern.
• Figure 3: Simulation results of the Initial Busting pattern in the AdEx model. The upper figure shows the variation of the membrane potential $V$ over time in the Initial Busting pattern, while the lower figure shows the variation of the adaptation current $\omega$ over time in the Initial Busting pattern.
• Figure 4: Simulation results of the Busting pattern in the AdEx model. The upper figure shows the variation of the membrane potential $V$ over time in the Busting pattern, while the lower figure shows the variation of the adaptation current $\omega$ over time in the Busting pattern.
• Figure 5: Simulation results of the Transient Spiking pattern in the AdEx model. The upper figure shows the variation of the membrane potential $V$ over time in the Transient Spiking pattern, while the lower figure shows the variation of the adaptation current $\omega$ over time in the Transient Spiking pattern.