Spike-Adding Mechanisms in the FitzHugh-Nagumo Model with Periodic Forcing: A Geometric Singular Perturbation Theory Approach

核心概念

This paper investigates the mechanisms behind spike-adding in the FitzHugh-Nagumo model with periodic forcing, revealing how the interplay of folded singularities and canard solutions contributes to the emergence and growth of spikes within a burst as the forcing frequency and amplitude vary.

摘要

Melland, P., Curtu, R., & Aminzare, Z. (2024). SPIKE-ADDING MECHANISMS IN A THREE-TIMESCALE FAST-SLOW SYSTEM: INSIGHTS FROM THE FITZHUGH-NAGUMO MODEL WITH PERIODIC FORCING. arXiv.

This paper aims to elucidate the spike-adding mechanism in a three-timescale FitzHugh-Nagumo (FHN) model driven by periodic input, focusing on the role of geometric singular perturbation theory (GSPT) and canard solutions in this process.

从中提取的关键见解

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

by Pake Melland... 在 arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00152.pdf

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How might the spike-adding mechanisms identified in this study generalize to other excitable systems beyond neuronal models?

The spike-adding mechanisms explored in this study, primarily revolving around folded singularities (folded nodes and saddles) in fast-slow systems, have the potential to generalize beyond neuronal models to a broader class of excitable systems. Here's why:

Ubiquity of Excitable Systems: Excitable systems are found in diverse fields beyond neuroscience, including cardiology, chemical kinetics, and laser physics. These systems share the characteristic of having a rest state from which they can be excited to produce a large-amplitude response (a spike or burst) when perturbed sufficiently.
Fast-Slow Dynamics: The mathematical framework of fast-slow analysis, which underpins the study of folded singularities, is applicable to systems exhibiting multiple timescales. This timescale separation is a common feature in many excitable systems, where processes like activation and recovery often operate at different rates.
Folded Singularities as Organizing Centers: Folded singularities act as organizing centers for complex dynamics in fast-slow systems. The presence of these singularities can lead to phenomena like canard explosions, where small changes in parameters cause dramatic shifts in the system's behavior, including the emergence of bursting oscillations and spike-adding transitions.
Therefore, the principles gleaned from studying spike-adding in neuronal models, particularly the role of folded singularities, could offer valuable insights into similar phenomena observed in other excitable systems. For instance, understanding how the interplay of timescales and folded singularities governs spike-adding could be relevant for:

Cardiac Arrhythmias: Investigating how changes in the heart's electrical activity, potentially influenced by external pacing or intrinsic factors, lead to irregular heartbeats.
Chemical Oscillations: Exploring how varying the concentrations of reactants or reaction rates in a chemical system can trigger transitions between different oscillatory regimes, including those exhibiting spike-adding.
Laser Dynamics: Analyzing how modulating parameters like pump power or cavity losses in lasers can induce changes in the pulse emission patterns, potentially leading to spike-adding behavior.
However, it's crucial to acknowledge that the specific details of spike-adding mechanisms might differ across systems due to variations in their underlying dynamics and the nature of the forcing or perturbations they experience.

Could noise, which is inherent in biological systems, disrupt the precise spike-adding patterns predicted by the deterministic model?

Yes, noise, an inherent feature of biological systems, can indeed disrupt the precise spike-adding patterns predicted by deterministic models like the one presented in the study. Here's how:

Deterministic Models as Approximations: Deterministic models provide a simplified representation of reality, neglecting the inherent randomness present in biological systems. In neurons, for example, ion channels open and close stochastically, synaptic inputs arrive irregularly, and there's background electrical noise.
Noise-Induced Transitions: Even small amounts of noise can have significant effects on systems near bifurcation points, which are crucial for spike-adding transitions. Noise can cause the system to prematurely switch between different dynamical regimes, leading to deviations from the predicted spike-adding patterns.
Altered Canard Trajectories: Noise can perturb the trajectories of canards, which are sensitive solutions that lie close to both attracting and repelling regions of the phase space. These perturbations can alter the timing and number of spikes within a burst, disrupting the precise spike-adding sequence.
Therefore, while deterministic models offer valuable insights into the potential mechanisms of spike-adding, it's essential to consider the role of noise in shaping neuronal activity. Incorporating noise into the model, perhaps through stochastic differential equations or numerical simulations with random perturbations, can provide a more realistic picture of how spike-adding might manifest in the presence of biological variability.

If we view the periodic forcing as a simplified representation of rhythmic brain activity, what implications do these findings have for understanding how different brain rhythms might influence neuronal firing patterns in vivo?

Viewing the periodic forcing as a simplified representation of rhythmic brain activity offers intriguing implications for understanding how different brain rhythms might influence neuronal firing patterns in vivo:

Brain Rhythms as Timing Cues: The study demonstrates that the frequency and amplitude of the periodic forcing significantly impact the number of spikes within a burst. This suggests that brain rhythms, which are oscillations in neuronal activity at various frequencies, could act as timing cues that modulate the firing patterns of individual neurons or neuronal populations.
Frequency-Dependent Modulation: The findings highlight the importance of the forcing frequency relative to the neuron's intrinsic timescales. This implies that different brain rhythms, characterized by distinct frequency bands (e.g., delta, theta, alpha), might exert distinct influences on neuronal firing. For instance, slower rhythms might be more effective at inducing bursting or spike-adding, while faster rhythms might have subtler modulatory effects.
Synchronization and Information Processing: The study's focus on spike-adding mechanisms suggests that brain rhythms could play a role in synchronizing the activity of neuronal ensembles. By coordinating the timing of spikes across multiple neurons, brain rhythms might facilitate information transfer and processing within and between brain regions.
However, it's crucial to acknowledge that the model used in the study is a simplification of the complex dynamics observed in the brain. Real brain rhythms are not perfectly periodic, and neurons are embedded in intricate networks with diverse inputs and connections. Nevertheless, the study provides a conceptual framework for investigating how rhythmic brain activity might shape neuronal firing patterns and contribute to cognitive functions. Further research, incorporating more realistic representations of brain rhythms and neuronal networks, is needed to fully elucidate the interplay between these oscillations and neuronal activity in vivo.