Six decades of the FitzHugh-Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines

TL;DR

The FitzHugh–Nagumo model provides a minimal yet powerful framework to study excitable dynamics, bridging neuroscience and wider physical and biological systems. The paper organizes the landscape into three layers: the original two-variable FHN, the diffusively coupled extension, and discretely coupled networks, each accompanied by stability and bifurcation analyses to map out fixed points, traveling waves, and pattern formation. Key contributions include analytic expressions for Hopf and saddle-node bifurcations, 1D and 2D Turing patterns, front interactions, traveling pulses, pacemaker-driven wave trains, and the emergence of chimera states in discrete networks. The work offers a practical, cross-disciplinary guide for modelers to harness FHN dynamics in complex, spatially extended systems with potential impacts on cardiac and neural modeling, pattern formation, and network theory.

Abstract

The FitzHugh-Nagumo equation, originally conceived in neuroscience during the 1960s, became a key model providing a simplified view of excitable neuron cell behavior. Its applicability, however, extends beyond neuroscience into fields like cardiac physiology, cell division, population dynamics, electronics, and other natural phenomena. In this review spanning six decades of research, we discuss the diverse spatio-temporal dynamical behaviors described by the FitzHugh-Nagumo equation. These include dynamics like bistability, oscillations, and excitability, but it also addresses more complex phenomena such as traveling waves and extended patterns in coupled systems. The review serves as a guide for modelers aiming to utilize the strengths of the FitzHugh-Nagumo model to capture generic dynamical behavior. It not only catalogs known dynamical states and bifurcations, but also extends previous studies by providing stability and bifurcation analyses for coupled spatial systems.

Figures (18)

• Figure 1: Exploring neuronal dynamics: The Hodgkin-Huxley model and its impact on subsequent research via the FitzHugh-Nagumo model.A. Depiction of the action potential across an axonal membrane, adapted from Hodgkin and Huxley's seminal work hodgkin1939action, with the potential difference measured in millivolts and the external environment set as the zero potential reference. B. Illustration of ion channels within a neuronal axon, detailing the exchange of sodium and potassium ions between the extracellular medium (EM) and the intracellular medium (IM). C. Demonstration of excitable behavior as characterized by the HH model, adapted from Fig. 17 from the original study HodgkinHuxley. D. Publication and citation data gathered from the Web of Science using "FitzHugh-Nagumo" as the search term (September 19, 2024). Note that these metrics do not include papers where the FHN model is not the primary focus, or papers published before the 1970s when the model name was first used. This shows the impact of the subsequent research using the FHN model as a simplified description of the HH model dynamics.
• Figure 2: Electrical analog of the FitzHugh-Nagumo modelA. Circuit representation: The FHN model's electrical counterpart includes key components that mimic biological neuronal dynamics. This circuit comprises: (i) A capacitor, symbolizing the neuronal membrane's capacitance $C$. (ii) A tunnel diode, representing the nonlinear ionic current $F(V)$. (iii) A resistor, indicative of the channel resistance $R$. (iv) An inductor $L$ and a battery $E$, completing the circuit to model the rest of the system's dynamics. B. Neuronal dynamics regimes: Through simulations of Eq. \ref{['EqTemporal']}, we capture essential neuronal behaviors: (i) Excitable regime: At $(a,b,\varepsilon)=(0.1,1.5,0.1)$, the circuit mimics the excitable nature of neurons, responding robustly to stimuli beyond a certain threshold. (ii) Oscillatory regime: With parameters $(a,b,\varepsilon)=(0,0.5,0.01)$, the system exhibits periodic oscillations, typical of active neuronal firing patterns. C. Spatial coupling in neuronal arrays: Extending the model to encompass spatial interactions involves linking multiple such circuits. This approach, as demonstrated in J. Nagumo et al.'s seminal work 2, allows for the exploration of wave propagation and collective behaviors in a network of neuron-like elements, mirroring the complex dynamics observed in biological neural networks.
• Figure 3: Overview of FitzHugh-Nagumo model applications This table organizes selected literature related to different formulations of the FitzHugh-Nagumo equations, as outlined in Eq. \ref{['EqTemporal']}, Eq. \ref{['EqSpatial']}, and Eq. \ref{['EqCoupled']}. Each listed study is categorized by its application domain and primary focus. It is important to note that, alongside the conventional FitzHugh-Nagumo models, certain studies incorporate variations like piecewise linear models or three-dimensional adaptations to meet particular investigative requirements.
• Figure 4: Fixed points in the FHN model.A. The nullclines in the ($u,v$) phase space ($\varepsilon=0.01$) and the effect of the parameters $a$ and $b$ on them are shown. B. The saddle node bifurcation in parameter space ($\varepsilon=0.01$) is shown and the mono- (1 FP) and tri-valued (3 FP) regions are differentiated. SN indicates the saddle node bifurcations and their intersection in the cusp (C) bifurcation.
• Figure 5: Dynamics and bifurcations of the FHN model for$\varepsilon=0.01$. A. Representative time series and phase space dynamics showing relaxation oscillations ($b=0.5$), excitability ($b=1.5$ and $a=0.1$), and bistability ($b=2$). B. Analytically derived Hopf and saddle node bifurcations, along with homoclinic bifurcations determined numerically, delineating various dynamical regions in the ($a,b$) parameter space. An expanded view of a specific area, marked by a black rectangle, is shown on the right. Black dots correspond to parameters used in A, and dashed lines mark sections analyzed in C. C Bifurcation diagrams of three representative regions. The diagram on the left (C.1) hints at an imminent homoclinic bifurcation due to the close proximity of the limit cycle to the saddle-node. The right diagram (C.3), on the other hand, reveals a shift from stability to instability, lacking an associated limit cycle, suggesting a global bifurcation has taken place. The extrema of limit cycle oscillations are shown by the solid gray line.