Spike-Adding Mechanisms in a Three-Timescale System: Insights from the FitzHugh-Nagumo Model with Periodic Forcing

TL;DR

This paper explores the spike-adding mechanism in three-timescale neuronal models with periodic forcing, revealing how input frequency and amplitude influence spike counts and characterizing the underlying geometric structures.

Contribution

It introduces a geometric singular perturbation approach to analyze spike-adding in forced FitzHugh-Nagumo and Morris-Lecar models, linking dynamics to brain rhythm frequencies.

Findings

Spike-adding diagram generated for FHN model.

Similar spike-adding structure observed in Morris-Lecar model.

Critical and super-critical manifolds characterized using geometric methods.

Abstract

In this work, we investigate the spike-adding mechanism in a class of three-dimensional fast-slow systems with three distinct timescales, inspired by the FitzHugh-Nagumo (FHN) model driven by periodic input. First, we numerically generate a spike-adding diagram for the FHN model by varying the frequency and amplitude of the input, revealing that as the frequency decreases and the amplitude increases, the number of spikes within each burst grows. We demonstrate that a similar spike-adding structure occurs in the more realistic, periodically forced Morris-Lecar neuronal model. Next, we apply methods from geometric singular perturbation theory to compute critical and super-critical manifolds of the fast-slow system. We use them to characterize the emergence of new burst-spikes in the FHN model, when the periodic forcing resembles a low frequency-band brain rhythm. We then describe how the uncovered spike-adding mechanism defines the boundaries that separate regions with different spike counts in the parameter space.