Bifurcations and Canards in the FitzHugh-Nagumo System: A Tutorial on Fast-Slow Dynamics with Application to Neuronal Excitability

Core Concepts

This article provides a tutorial on the dynamics of the FitzHugh-Nagumo system, a simplified model of neuronal excitability, using the framework of fast-slow dynamical systems and bifurcation theory.

Abstract

This article investigates the FitzHugh-Nagumo (FH-N) system, a simplified model derived from the Hodgkin-Huxley model, to understand the dynamics of neuronal excitability.

Introduction and Background

• The article begins by highlighting the oscillatory and excitability dynamics observed in neuronal activity and cardiac rhythm.
• It introduces the Hodgkin-Huxley (HH) model, a complex system of four differential equations that describe the propagation of electrical signals along a squid's giant axon.
• The FH-N model is presented as a simplified two-variable version of the HH model, aiming to make the dynamics of excitability more understandable.

Fast-Slow Dynamics and the FitzHugh-Nagumo System

• The article introduces the concept of fast-slow systems, where some variables evolve at significantly different rates than others.
• It explains how the FH-N model can be transformed into a fast-slow system by appropriate time scaling.
• The critical manifold, defined by setting the fast variable's derivative to zero, is introduced as a crucial element in analyzing fast-slow systems.

The Singular Case (ε = 0)

• The singular case, where the time scale separation parameter ε is set to zero, is analyzed to gain insights into the system's behavior.
• The dynamics of the system are explored for different parameter values, leading to various scenarios involving stable and unstable equilibria, limit cycles, and trajectories tending to infinity.
• The period of the limit cycle, representing oscillatory behavior, is estimated based on the time spent by the trajectory on different branches of the critical manifold.

The Regular Case (0 < ε ≪1)

• Fenichel's theorem is introduced to connect the dynamics of the singular case (ε = 0) to the regular case (0 < ε ≪1).
• The theorem guarantees the existence of a slow manifold, close to the critical manifold, where the system's trajectories evolve slowly.
• The article provides a method to approximate the slow manifold and discusses its implications for understanding the system's behavior for small but non-zero ε.

Bifurcations

• The article investigates the influence of parameters on the system's dynamics by analyzing bifurcations.
• Two specific cases are considered: one with varying parameter c and the other with varying parameter b.
• Hopf bifurcations, leading to the emergence of limit cycles, are identified and analyzed for both cases.
• The stability of the equilibrium points and limit cycles is discussed in relation to the parameter values.

Conclusion

• The article provides a comprehensive tutorial on the dynamics of the FH-N system using the framework of fast-slow systems and bifurcation theory.
• It highlights the importance of understanding the interplay between fast and slow variables in shaping the system's behavior.
• The insights gained from this analysis contribute to a deeper understanding of neuronal excitability and can be extended to other biological systems exhibiting similar dynamics.

Stats

The period of the limit cycle for b = c = 0 is approximately 6.45 time units.
For b < 0, the FitzHugh-Nagumo system has one unstable node/focus equilibrium and two saddle equilibria.
For 0 < b ≤ 1/4, the system has a stable limit cycle and the origin is an unstable node/focus equilibrium.
The Hopf bifurcation occurs at bHε = (-4 + √(16 + 3ε))/ε for 0 < ε ≪ 1.
As ε approaches 0, bHε approaches 3/8.

Quotes

"Our goal with the present article is to exhibit the various dynamics in the multiple time scale model and motivate their existence in the light of methods from geometric singular perturbation theory and bifurcation theory."
"All figures in this article were created through numerical simulations conducted in Matlab, using integration functions such as ode15s or ode23s, and, in more sensitive examples, the MatCont toolbox developed by Willy Govaerts, Yuri A. Kuznetsov and Hil G.E. Meijer [7]."

Key Insights Distilled From

Bifurcations and canards in the FitzHugh-Nagumo system: a tutorial in fast-slow dynamics

by Brun... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11209.pdf

Deeper Inquiries

How can the analysis of fast-slow dynamics in the FitzHugh-Nagumo model be extended to understand more complex neuronal behaviors, such as bursting or synchronization?

The FitzHugh-Nagumo (FHN) model, with its ability to capture the basic excitability of neurons, serves as a stepping stone to understanding more intricate neuronal dynamics. Here's how the analysis can be extended to understand bursting and synchronization:
1. Bursting:

Model Modification: The standard FHN model can be modified to incorporate additional slow variables or time scales to represent the slow processes underlying bursting. For instance:

Adding a slow current: Introducing a slow potassium current that activates during spiking and deactivates slowly can lead to periods of spiking followed by quiescence, characteristic of bursting.
Coupling to metabolic processes: Coupling the FHN model to equations representing slow metabolic processes that modulate neuronal excitability can also generate bursting patterns.

Analysis Techniques:

Fast-slow analysis: Similar to the tutorial, identifying the critical manifold and its bifurcations in the extended model is crucial. Bursting often arises from the interplay between slow passage through bifurcations and fast dynamics on different branches of the critical manifold.
Phase plane analysis: Visualizing trajectories in the phase space of the extended model can provide insights into the mechanisms of burst initiation and termination.
2. Synchronization:

Network Coupling: To study synchronization, multiple FHN neurons can be coupled together in a network. The coupling can be through:

Electrical synapses: Directly connecting the membrane potentials of neighboring neurons.
Chemical synapses: Modeling the release and reception of neurotransmitters between neurons.

Analysis Techniques:

Phase response curves: These curves characterize how the timing of a neuron's spike is affected by external inputs. Analyzing the phase response curves of coupled neurons can predict synchronization properties.
Network dynamics: Studying the collective behavior of the coupled FHN neurons, such as the emergence of synchronized oscillations or traveling waves, can shed light on network-level phenomena.
Key Considerations:

Biological plausibility: When extending the FHN model, it's essential to ensure that the modifications and parameters used are biologically realistic.
Computational complexity: Analyzing complex neuronal behaviors often requires numerical simulations and sophisticated computational tools.
By combining these approaches, the analysis of fast-slow dynamics in extended FHN models can provide valuable insights into the mechanisms underlying bursting, synchronization, and other complex neuronal phenomena.

Could the presence of noise, inherent in biological systems, significantly alter the dynamics and bifurcations observed in the deterministic FitzHugh-Nagumo model?

Yes, the presence of noise, a ubiquitous feature of biological systems, can significantly alter the dynamics and bifurcations observed in the deterministic FitzHugh-Nagumo (FHN) model. Here's how:
1. Noise-Induced Phenomena:

Stochastic Resonance: Noise can enhance the system's response to weak, subthreshold signals by pushing the system across a bifurcation point it wouldn't cross deterministically. This phenomenon is known as stochastic resonance.
Noise-Induced Transitions: Noise can induce transitions between different attractors of the system, even if these transitions are not possible in the deterministic case. For example, noise can cause a neuron to fire even if the input is below the deterministic threshold.
Coherence Resonance: In some cases, noise can actually enhance the regularity of oscillations in the FHN model, leading to more coherent spiking behavior.
2. Modification of Bifurcations:

Bifurcation Delay: Noise can delay the onset of bifurcations, meaning that the system might remain in a particular dynamical regime for a wider range of parameters than predicted by the deterministic model.
Bifurcation Shifting: Noise can shift the location of bifurcation points, altering the parameter values at which transitions between different dynamical regimes occur.
New Bifurcations: In some cases, noise can even induce new types of bifurcations that are not present in the deterministic model.
3. Modeling Noise:

Additive Noise: This type of noise is typically modeled by adding a random term to the equations of the FHN model, representing fluctuations in the membrane potential or other variables.
Multiplicative Noise: This type of noise is modeled by multiplying a variable in the FHN model by a random term, representing fluctuations in parameters like ion channel conductances.
Implications for Neuronal Dynamics:

Robustness and Variability: Noise can both enhance the robustness of neuronal responses to perturbations and contribute to the variability observed in neuronal firing patterns.
Information Processing: Noise can play a constructive role in neuronal information processing by expanding the range of dynamical responses and enhancing sensitivity to weak signals.
Key Takeaway:
Understanding the interplay between deterministic dynamics and noise is crucial for accurately modeling and interpreting neuronal behavior. Stochastic versions of the FHN model, incorporating noise terms, provide a more realistic representation of neuronal dynamics in biological settings.

What are the broader implications of understanding fast-slow dynamics and bifurcations in biological systems beyond neuronal excitability?

The principles of fast-slow dynamics and bifurcations extend far beyond neuronal excitability, offering a powerful framework for understanding a wide range of phenomena in biological systems. Here are some broader implications:
1. Cellular Processes:

Cell Cycle Regulation: The cell cycle, a fundamental process of cell division and growth, involves a sequence of events governed by complex regulatory networks. Fast-slow dynamics and bifurcations can explain transitions between different phases of the cell cycle, such as the switch from growth to DNA replication.
Metabolic Oscillations: Many metabolic pathways exhibit oscillations, with concentrations of metabolites fluctuating over time. These oscillations can be understood as arising from fast-slow interactions between enzymatic reactions and slower regulatory mechanisms.
Signal Transduction: Cells communicate through intricate signaling pathways that transmit information from the cell surface to the nucleus. Fast-slow dynamics and bifurcations can explain how cells respond to external signals in a switch-like or oscillatory manner.
2. Organismal Level:

Cardiac Rhythms: The beating of the heart relies on the coordinated electrical activity of cardiac cells. Fast-slow dynamics and bifurcations are essential for understanding normal heart rhythms and how disruptions in these dynamics can lead to arrhythmias.
Hormonal Regulation: The endocrine system controls physiological processes through the release of hormones. Fast-slow dynamics and bifurcations can explain how hormone levels are regulated and how they fluctuate in response to internal and external cues.
Population Dynamics: Predator-prey interactions, disease spread, and other ecological phenomena can be modeled using systems of differential equations. Fast-slow dynamics and bifurcations can explain oscillations, thresholds, and other complex patterns observed in population dynamics.
3. Synthetic Biology and Bioengineering:

Design Principles: Understanding fast-slow dynamics and bifurcations provides design principles for engineering synthetic biological systems with desired dynamical behaviors, such as oscillators, switches, and logic gates.
Control Strategies: By manipulating parameters that influence fast-slow dynamics and bifurcations, it may be possible to control biological systems for therapeutic purposes, such as suppressing pathological oscillations or inducing desired cellular responses.
Key Takeaway:
The concepts of fast-slow dynamics and bifurcations provide a unifying framework for understanding how complex behaviors emerge from the interplay of processes occurring at different time scales in biological systems. This understanding has implications for basic research, disease modeling, drug discovery, and synthetic biology.