Spike-adding Mechanisms in a Three-timescale System: Insights from the Fitzhugh-Nagumo Model with Periodic Forcing

Pake Melland; Rodica Curtu; Zahra Aminzare

doi:10.1137/24M1706980

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Spike-adding Mechanisms in a Three-timescale System: Insights from the Fitzhugh-Nagumo Model with Periodic Forcing

Journal article

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Spike-adding Mechanisms in a Three-timescale System: Insights from the Fitzhugh-Nagumo Model with Periodic Forcing

Pake Melland, Rodica Curtu and Zahra Aminzare

SIAM journal on applied dynamical systems, Vol.24(4), pp.2977-3011

01/01/2025

DOI: 10.1137/24M1706980

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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 supercritical 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.

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Details

Title: Subtitle: Spike-adding Mechanisms in a Three-timescale System: Insights from the Fitzhugh-Nagumo Model with Periodic Forcing
Creators: Pake Melland - Oregon Institute of Technology
Rodica Curtu - Univ Iowa, Dept Math, Iowa City, IA 52242 USA
Zahra Aminzare - University of Iowa
Resource Type: Journal article
Publication Details: SIAM journal on applied dynamical systems, Vol.24(4), pp.2977-3011
DOI: 10.1137/24M1706980
ISSN: 1536-0040
eISSN: 1536-0040
Publisher: Siam Publications
Number of pages: 35
Grant note: 2037828 / NSF; National Science Foundation (NSF) Stanley-UI Foundation Support Organization 1840260 / NSF-RTG; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS) MPS-TSM-00008005 / Simons Foundation
Language: English
Date published: 01/01/2025
Academic Unit: Iowa Neuroscience Institute; Mathematics
Record Identifier: 9985147199402771

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