FitzHugh-Nagumo Model

The FitzHugh-Nagumo model is a simplified mathematical model used to describe the excitation and propagation of electrical impulses in neurons and other excitable biological tissues, serving as a conceptual bridge between highly abstract neural models and more complex, biophysically detailed representations [1][8]. Classified as a two-dimensional simplification of the Hodgkin-Huxley equations, it is a cornerstone of theoretical neuroscience and nonlinear dynamics, designed to capture the essential qualitative features of neuronal spiking and recovery while remaining analytically and computationally tractable [7]. Its importance lies in providing a framework to understand the fundamental principles of excitability, oscillations, and wave propagation in neural systems without the complexity of modeling every ionic current [6]. The model operates by reducing the four-dimensional Hodgkin-Huxley system to two key variables: one representing a fast voltage-like variable for excitation and a second representing a slow recovery variable [8]. This abstraction retains the core "all-or-none" character of neural firing—a fundamental property noted in early computational neuroscience [4]—and produces a characteristic trajectory in phase space involving a stable resting state, threshold for excitation, and a cyclic spiking limit cycle. Key characteristics include its ability to demonstrate excitability, wherein a super-threshold perturbation triggers a large action-potential-like spike, and its capacity to support traveling pulse waves, making it useful for studying signal propagation. While the original model is its main type, numerous variations exist, often modifying the nonlinear functions to study different dynamical regimes or to incorporate additional biological features. The FitzHugh-Nagumo model finds significant application in exploring the dynamics of neural networks, pattern formation in cortical tissues, and the theoretical basis for cardiac arrhythmias. It has been instrumental in studies ranging from the analysis of rhythmic activity in central pattern generators, relevant for motor functions like walking [3], to the investigation of how temperature or pharmacological agents like tetrodotoxin affect excitability [2]. Its simplicity allows researchers to gain insights into how collective behaviors emerge from interconnected excitable units. In modern neuroscience, while the focus has shifted toward population-level approaches and more detailed cellular explanations [5], the FitzHugh-Nagumo model remains a vital pedagogical and theoretical tool for conceptualizing the nonlinear dynamics inherent in neural activity, illustrating that a useful model is an aid to understanding reality rather than a facsimile of it [7].

Overview

The FitzHugh-Nagumo (FHN) model is a simplified mathematical representation of neuronal excitability and action potential generation, serving as a canonical example of a two-dimensional excitable system. It was developed independently by Richard FitzHugh in 1961 and Jin-Ichi Nagumo et al. in 1962 as a reduction of the more physiologically detailed Hodgkin-Huxley equations [14]. The model's primary purpose is not to replicate the full biophysical complexity of a neuron but to capture the essential qualitative dynamics of excitation and recovery with minimal variables, thereby acting as an aid for understanding core principles of neural signaling [13]. It abstracts the four-dimensional Hodgkin-Huxley system into two key variables: a fast activation or excitation variable (often denoted v), analogous to the membrane potential and sodium channel activation, and a slow recovery variable (often denoted w), representing the combined effects of potassium channel activation and sodium channel inactivation [14].

Mathematical Formulation and Core Dynamics

The standard form of the FitzHugh-Nagumo model is described by a pair of coupled nonlinear ordinary differential equations:

$$\begin{aligned} \frac{dv}{dt} &= v - \frac{v^3}{3} - w + I_{\text{ext}} \\ \tau \frac{dw}{dt} &= v + a - b w \end{aligned}$$

Here, v represents the excitatory variable (membrane potential-like), and w is the inhibitory recovery variable. The parameter $I_{\text{ext}}$ denotes an externally applied current stimulus. The time constant $\tau$ (where $\tau \gg 1$) governs the slow evolution of the recovery variable relative to the fast excitation variable. The parameters a and b are constants that shape the model's nullclines and determine its resting state and excitability threshold [14]. The cubic function $f(v) = v - v^3/3$ in the dv/dt equation provides the essential nonlinearity that allows for regenerative excitation. It creates an N-shaped v-nullcline. The linear function in the dw/dt equation yields a straight-line w-nullcline. The intersection points of these nullclines define the system's fixed points. For a typical parameter set (e.g., $a = 0.7, b = 0.8, \tau = 12.5$), the system has a single stable fixed point representing the resting state [14]. A sufficiently large, brief perturbation of v (simulating an injected current pulse) can push the system past a separatrix or threshold, triggering a large, stereotyped excursion in phase space—a limit cycle representing an action potential—before returning to rest. This encapsulates the all-or-none firing property of neurons.

Relationship to Biological Neuron Models and Synaptic Integration

While abstract, the FHN model's variables and parameters can be loosely mapped to biophysical processes. The variable v integrates influences from fast, self-amplifying sodium currents (positive feedback) and the slower restorative forces encapsulated in w [14]. In more detailed biological neuron models, such as compartmental models that include branching dendritic trees, synaptic inputs are integrated across complex spatial structures, with voltage-dependent channels influencing local computations [14]. The FHN model strips away this morphological and channel diversity to focus on the temporal interaction of excitation and recovery at a single point, such as an axon initial segment or a simplified neuronal compartment. The model's response to sustained $I_{\text{ext}}$ illustrates fundamental firing modes. For subthreshold currents, the fixed point is stable. As $I_{\text{ext}}$ increases, the fixed point can lose stability via a Hopf bifurcation, leading to repetitive spiking (a stable limit cycle). With further increase, the system may enter a depolarization block where the fixed point becomes stable again. This progression qualitatively mirrors the behavior of real neurons. The model's phase plane analysis clearly illustrates the concepts of threshold, excitability, refractoriness, and anode-break excitation. The recovery variable w acts as a slow negative feedback, first allowing the spike to terminate and then imposing a relative refractory period where a larger stimulus is required to elicit a subsequent spike [14].

Role as a Pedagogical and Theoretical Tool

The FitzHugh-Nagumo model exemplifies the principle that a useful model should not be a facsimile of reality; it is an aid for understanding it [13]. Its simplicity makes it analytically and computationally tractable, allowing researchers to study phenomena like:

• The genesis of action potentials
• Phase resetting and synchronization of oscillators
• Wave propagation in spatially extended systems (e.g., the FHN cable equation)
• Bifurcations in neural dynamics

It serves as a foundational model in theoretical neuroscience and nonlinear dynamics, bridging highly simplified integrate-and-fire models and complex, biophysically detailed multi-compartmental models. Its two-dimensional nature permits complete visualization of trajectories in the phase plane (v-w space), providing intuitive insight into dynamics that are obscured in higher-dimensional systems like the Hodgkin-Huxley model [14]. Furthermore, variations of the model form the basis for studying kinetic models of synaptic transmission, where simplified neuronal elements are connected to explore network dynamics, pattern formation, and the conditions for stable rhythm generation in central pattern generators. In summary, the FitzHugh-Nagumo model is a seminal reduction that captures the quintessential features of neuronal excitability and oscillation through the minimal interaction of a fast activator and a slow inhibitor. It remains a cornerstone for teaching and theoretical analysis, providing a conceptual framework for understanding how nonlinear interactions between currents produce the complex electrical behavior of nerve cells [13][14].

History

The FitzHugh-Nagumo (FHN) model, a seminal simplification of the Hodgkin-Huxley equations for neuronal excitability, emerged from a confluence of mid-20th century advances in electrophysiology and mathematical biology. Its development represents a pivotal effort to distill the essential dynamics of nerve impulse generation into a mathematically tractable form, bridging detailed biophysical description and abstract dynamical systems theory.

Biophysical Foundations and Precursor Models (1939-1952)

The historical trajectory leading to the FHN model is inextricably linked to the quantification of ionic mechanisms underlying the action potential. Critical groundwork was laid by the pioneering voltage-clamp experiments of Kenneth S. Cole, who developed the technique in the 1930s, and its subsequent application by Alan L. Hodgkin and Andrew F. Huxley. Their seminal 1952 series of papers, most notably "Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo," provided the first complete mathematical description of action potential generation [15]. The Hodgkin-Huxley (HH) model was a system of four nonlinear differential equations that accurately reproduced the squid giant axon's electrical behavior by modeling the time- and voltage-dependent conductances of sodium and potassium channels. While profoundly successful, the HH model's complexity—with four state variables and numerous fitted parameters—made analytical treatment difficult and motivated the search for simpler, more conceptually transparent representations.

Simplification and the Bonhoeffer-van der Pol Analogy (1950s)

Parallel to the work in physiology, insights from nonlinear dynamics and engineering began to influence theoretical neurobiology. In the 1950s, the German chemist and biophysicist Karl Friedrich Bonhoeffer drew an explicit analogy between neural excitability and the behavior of a simple electrical circuit incorporating a tunnel diode, which exhibits negative resistance. He noted that this circuit, mathematically described by a modified van der Pol oscillator equation, could produce pulse-like responses similar to action potentials. The van der Pol oscillator, originally formulated by Balthasar van der Pol in the 1920s to model oscillations in electrical circuits using vacuum tubes, is a second-order nonlinear differential equation known for its relaxation oscillations. Bonhoeffer's conceptual work established the critical link: the excitable membrane could be viewed as an active circuit element with a nonlinear current-voltage relationship, and its dynamics could be captured by a system with just two key variables—one fast (akin to membrane voltage) and one slow (representing a recovery variable).

Formulation of the FitzHugh-Nagumo Equations (1961-1962)

Building directly upon Bonhoeffer's analogy and motivated by the complexity of the HH equations, the American biophysicist Richard FitzHugh systematically sought a minimal model. In his 1961 paper, "Impulses and Physiological States in Theoretical Models of Nerve Membrane," FitzHugh proposed a simplified two-variable model. He demonstrated that by applying a series of mathematical reductions to the HH equations—including phase-plane analysis and the identification of fast and slow manifolds—one could derive a system that retained the core qualitative behaviors: excitability, threshold behavior, and the existence of refractory periods. The model equations he presented were:

$$\epsilon \frac{dv}{dt} = v - \frac{v^3}{3} - w + I_{\text{ext}}, \quad \frac{dw}{dt} = v + a - b w$$

where $v$ represents a dimensionless membrane potential, $w$ is a recovery variable, $I_{\text{ext}}$ is an applied current, $\epsilon \ll 1$ ensures $v$ is fast, and $a$, $b$ are parameters [15]. Concurrently and independently, the Japanese electrical engineer Jin-ichi Nagumo, along with his colleagues Arimoto and Yoshizawa, constructed an equivalent electronic circuit in 1962. Their "Nagumo circuit" used a tunnel diode for the nonlinear negative resistance and an RC network to provide recovery, providing a physical instantiation of the equations. Nagumo's work solidified the model's engineering relevance and its name; it is thus universally known as the FitzHugh-Nagumo model.

Mathematical Analysis and Canonical Status (1960s-1980s)

Following its formulation, the FHN model became a cornerstone for the mathematical analysis of excitability. Its two-dimensional phase plane allowed for complete geometric analysis using nullclines, fixed points, and limit cycles. As noted earlier, the transition from a stable fixed point to a stable limit cycle via a Hopf bifurcation with increasing $I_{\text{ext}}$ provided a clear mathematical mechanism for the onset of repetitive spiking. The model's cubic $v$-nullcline (representing the fast current-voltage relation) and linear $w$-nullcline (representing slow recovery) became the canonical picture for excitability in dynamical systems textbooks. Throughout the 1970s and 1980s, it was extensively used to study wave propagation in one-dimensional cables (as a continuous PDE version) and in two-dimensional media to model re-entrant cardiac arrhythmias and spiral wave formation in chemical systems, demonstrating its broad applicability beyond neuroscience.

Extensions and Integration into Network Theories (1990s-Present)

The late 20th and early 21st centuries saw the FHN model evolve from a model of a single cell to a fundamental building block for network dynamics. It became a standard "processing unit" in studies of coupled oscillator networks, synchronization, and pattern formation. Its simplicity made it ideal for large-scale simulations of neural populations. Furthermore, its conceptual framework directly influenced the development of other simplified neuronal models, such as the Morris-Lecar model (which incorporates more biophysical detail while retaining two dimensions) and the Izhikevich model (which combines quadratic nonlinearity with an adaptive variable). The model's principles also informed the analysis of more complex, non-Markovian stochastic systems used to describe biological neural nets. For instance, studies have considered infinite systems of interacting components with memory of variable length, where the fundamental excitable dynamics of individual units, as captured by FHN-like elements, interact through complex, history-dependent coupling [16]. This aligns with broader conceptual advances suggesting that traditional selectivity principles in neural coding must be revised to account for network-level emergent phenomena and history-dependent plasticity, where simplified models like FHN provide the essential substrate for such interactions.

Contemporary Relevance and Computational Role

Today, the FitzHugh-Nagumo model remains a vital tool in theoretical neuroscience, computational biology, and nonlinear dynamics. It serves as a primary educational exemplar for teaching bifurcation theory and neural dynamics. In computational research, it is frequently employed as a testbed for new numerical methods for solving differential equations and for studying the collective behavior of large-scale neural simulations where the computational cost of full HH models is prohibitive. Its legacy persists as the archetypal example of how mathematical abstraction can yield profound insight into complex biological function, encapsulating the essence of excitability in a form that is both analytically accessible and rich in dynamical behavior.

Description

The FitzHugh-Nagumo (FHN) model is a simplified, two-dimensional mathematical representation of neuronal excitability and action potential generation. It serves as a canonical example of a relaxation oscillator and provides a qualitative approximation of the dynamics described by the more complex Hodgkin-Huxley equations. The model reduces the essential biophysical processes to two key variables: a fast voltage-like variable, $v$, representing membrane potential and activation, and a slow recovery variable, $w$, representing inactivation or refractoriness [6]. Its development marked a significant step in the application of dynamical systems theory to neuroscience, offering a tractable framework for analyzing excitability, oscillations, and bifurcations in neural systems without the computational burden of detailed biophysical models [13].

Mathematical Formulation and Biophysical Analogy

The standard form of the FitzHugh-Nagumo equations is:

$$\begin{aligned} \frac{dv}{dt} &= v - \frac{v^3}{3} - w + I_{\text{ext}}, \\ \frac{dw}{dt} &= \epsilon (v + a - b w). \end{aligned}$$

Here, $I_{\text{ext}}$ represents an externally applied current, $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$) that ensures $w$ evolves much more slowly than $v$, and $a$ and $b$ are constants that shape the model's nullclines and dynamics [13]. The cubic nonlinearity in the $v$-equation ($v - v^3/3$) provides a simplified representation of the combined, rapid activation and inactivation of voltage-gated sodium channels that underlies the action potential's upstroke and downstroke [2]. The linear recovery term involving $w$ approximates the slower dynamics of potassium channel rectification and sodium channel inactivation that are responsible for the refractory period [2][6]. This formulation abstracts the fundamental ionic mechanisms established in seminal work on the squid giant axon, where currents carried by sodium and potassium ions through the axonal membrane were precisely characterized [2]. The model captures the all-or-nothing nature of spike generation and the existence of a distinct threshold, albeit in a smoothed, mathematically continuous form compared to the sharp threshold in biophysically detailed models. The separation of timescales enforced by $\epsilon$ is crucial for generating the characteristic excitable behavior and relaxation oscillations, mirroring the fast depolarization and slow repolarization phases of a biological action potential [13].

Phase Plane Analysis and Excitability

Analysis of the model in the $(v, w)$ phase plane reveals its core behaviors. The system's fixed points are determined by the intersections of the $v$-nullcline ($w = v - v^3/3 + I_{\text{ext}}$), which is cubic, and the $w$-nullcline ($w = (v + a)/b$), which is linear [13]. For a range of parameters, the system possesses a single stable fixed point. A sufficiently small, brief perturbation from this rest state will result in a small excursion in phase space before the system returns directly to the fixed point, representing a subthreshold response. However, a perturbation that crosses a separatrix in the phase plane triggers a large-amplitude excursion—a loop around the phase plane—before returning to rest. This large excursion models the generation of a single action potential [13]. The excitability of the model is thus a geometric property of its phase portrait. The recovery variable $w$ acts as a slow negative feedback: during the spike's upstroke, $v$ increases rapidly while $w$ changes little; as $v$ peaks, the increase in $w$ pulls the trajectory back toward the resting state, creating the downstroke and subsequent refractory period during which the neuron is less excitable. This interplay creates the characteristic shape of the action potential and the property of refractoriness. As noted earlier, with a sustained increase in $I_{\text{ext}}$, the fixed point can lose stability, leading to repetitive spiking.

Relation to Detailed Biophysics and Conceptual Shifts

The FHN model exemplifies a middle ground in the spectrum of neuronal modeling, situated between highly abstract integrate-and-fire models and meticulously detailed multi-compartmental models that include explicit channel kinetics, dendritic arborization, and synaptic integration [13][17]. It sacrifices biophysical detail for analytical tractability, yet retains enough structure to qualitatively reproduce key phenomena like excitability, anode-break excitation, and post-inhibitory rebound. This approach aligns with a reductionist methodology that has dominated neuroscience, seeking to explain higher-level functions (like signal transmission) through properties at lower organizational levels (like membrane currents) [6]. However, the utility of such simplified models persists even as computational power enables extraordinarily detailed simulations. They provide fundamental insights into the types of dynamics possible in neural systems and serve as building blocks for network models where simulating thousands of detailed Hodgkin-Huxley neurons remains prohibitive [13]. Furthermore, the principles encapsulated in the FHN model find relevance in modern neuroengineering interventions. For instance, closed-loop neuromodulation systems, such as brain-spine interfaces that decode motor intent to modulate epidural electrical stimulation, ultimately rely on manipulating the collective excitability and oscillatory dynamics of neuronal populations—dynamics for which the FHN model provides a foundational conceptual language [3]. The model also implicitly relates to structural components of neurons. The point-neuron assumption of FHN abstracts away the spatial extent of the neuron. In biological neurons, the initiation of an action potential typically occurs at the axon initial segment, and its propagation along unmyelinated axons relies on local circuits of ionic current flowing into active membrane regions and out through adjacent sections [18]. Myelination alters this conduction dramatically, but the core excitability mechanism at each node of Ranvier remains analogous to the local dynamics described by models like FHN [18][19]. Similarly, while the model does not include dendritic computation, the integration of synaptic inputs that lead to the threshold crossing represented in the model occurs across the dendritic tree, which is involved in complex signaling and even local protein synthesis [17]. Building on the concept discussed above regarding its educational role, the FHN model's simplicity makes it an ideal vehicle for exploring how collective neural dynamics emerge from individual unit properties. This is pertinent to contemporary shifts in neuroscience, where population-level approaches and the revision of traditional principles like "grandmother cell" selectivity are emphasizing the distributed nature of coding [5]. A network of coupled FHN oscillators can exhibit synchronization, waves, and pattern formation, illustrating how basic excitable units can give rise to complex system-level behaviors relevant to central pattern generators or cortical oscillations [13]. Thus, while rooted in mid-20th century reductionism, the FitzHugh-Nagumo model continues to be a vital tool for connecting cellular-level mechanisms to circuit and systems-level functions [6][13].

Significance

The FitzHugh-Nagumo (FHN) model occupies a pivotal position in theoretical neuroscience and nonlinear dynamics, extending far beyond its origins as a simplified representation of the Hodgkin-Huxley equations. Its mathematical elegance and computational tractability have made it a foundational tool for exploring fundamental principles of excitability, wave propagation, and pattern formation across diverse biological systems. While the model's core dynamics, including the transition to repetitive spiking via a Hopf bifurcation with increased external current, have been established in prior sections, its significance is amplified by its broad applicability to excitable tissues beyond neurons and its role in modern computational paradigms.

A General Model for Cellular Excitability

Although neuronal action potentials are its most familiar context, the excitable dynamics captured by the FHN equations are ubiquitous in physiology [7]. The model's variables—a fast activation variable (v) and a slow recovery variable (w)—effectively abstract the interplay between rapid depolarizing currents and slower restorative processes. This abstraction proves directly relevant to cardiac muscle cells, where similar excitation-recovery cycles govern the coordinated contractions of the heart, and to certain endocrine cells that exhibit regenerative electrical activity to control hormone release [7]. The model's ability to reproduce phenomena like refractoriness and threshold behavior without the complexity of detailed ion-channel kinetics makes it an indispensable tool for studying wave propagation in cardiac tissue and spiral wave formation, which is analogous to life-threatening cardiac arrhythmias.

Application to Specialized Sensory Neurons

The utility of the FHN framework is particularly evident in the study of specialized neuronal subclasses, such as somatosensory cold thermoreceptors. These neurons transduce temperature decreases into patterned action potential discharges. Modeling their activity requires capturing adaptation and specific firing patterns in response to sustained stimuli. The FHN model, often with appropriate parameter modifications or coupling to additional slow variables, provides a scaffold for investigating how the intrinsic excitability of these cells shapes the neural code for temperature. By simulating the response of an FHN-type cold thermoreceptor to a cooling ramp, researchers can bridge cellular properties to perceptual phenomena, such as the perception of cold intensity and adaptation.

Foundation for Modern Neuromorphic and Machine Learning Systems

Beyond biological modeling, the FHN model's simplicity has catalyzed advances in neuromorphic engineering and machine learning. Its formulation is highly amenable to implementation in analog and digital very-large-scale integration (VLSI) circuits, serving as the computational unit in hardware-based spiking neural networks (SNNs). These networks leverage the temporal dynamics of spikes for efficient, brain-inspired computation. Notably, FHN-like neuron models are employed in the implementation of recurrent spiking neural networks (RSNNs) that can learn through hardware-friendly algorithms like e-prop (eligibility propagation) [19]. E-prop enables online learning in networks of spiking neurons by approximating gradient descent without the memory overhead of backpropagation through time, making it suitable for low-power neuromorphic chips. The FHN model, with its clear separation of timescales, offers a tractable substrate for developing and testing such learning rules in silico.

A Pedagogical and Analytical Cornerstone

As noted earlier, the model serves as a primary educational exemplar. Its two-dimensional phase plane allows for complete geometric analysis using nullclines, vector fields, and bifurcation diagrams, making it the simplest model that exhibits the full repertoire of neural excitability: a stable resting state, threshold behavior, all-or-nothing action potentials, and refractoriness [7]. Students can visually grasp how the intersection of the v-nullcline (a cubic function) and the w-nullcline (a linear function) determines fixed points, and how their stability changes with parameters like the external current $I_{\text{ext}}$ [22]. This analytical transparency is why the FHN model is frequently the first system used to teach concepts like:

• Phase plane analysis and stability of fixed points
• The existence and stability of limit cycles
• Separatrix and threshold phenomena
• Hopf and saddle-node bifurcations in a biological context

Bridging Phenomenology and Biophysical Detail

The FHN model operates at a crucial intermediate level of description. It is more biophysically grounded than purely phenomenological integrate-and-fire models, as its variables are loosely analogous to membrane voltage and a composite recovery current. Yet, it is far simpler than the full Hodgkin-Huxley formalism, which tracks multiple gating variables and ion-specific conductances [20][22]. This middle ground allows researchers to incorporate essential biological features without intractable complexity. For instance, synaptic input can be added as a current term $I_{\text{syn}}$ in the voltage equation, where conductance-based synapses would follow $I_{\text{syn}} = g_{\text{syn}} (t) (v - E_{\text{syn}})$, with $g_{\text{syn}}$ representing the time-varying synaptic conductance and $E_{\text{syn}}$ the reversal potential [19]. The postsynaptic integration of such signals, determining whether the model neuron will fire, mirrors the biological process where a neuron sums excitatory and inhibitory inputs [19]. Furthermore, the model can be extended to explore the effects of morphology by coupling multiple FHN units along a cable-like structure, providing insights into signal propagation in non-myelinated fibers or within dendritic arbors, which come in varied forms such as multipolar or bipolar arrangements [17]. In summary, the significance of the FitzHugh-Nagumo model is multifaceted. It provides a universal mathematical language for excitability applicable from neurons to cardiac cells, enables the study of specialized sensory transduction, forms the basis for next-generation neuromorphic computing architectures, and offers an unparalleled educational tool for nonlinear dynamics. By distilling the essence of excitation and recovery into a minimal yet powerful pair of equations, it continues to bridge disciplines, from theoretical biology and physics to computer engineering and mathematics, ensuring its enduring role as a cornerstone of complex systems science.

Applications and Uses

The FitzHugh-Nagumo (FHN) model, while originally conceived as a simplified representation of neuronal action potentials, has found extensive utility across multiple scientific and engineering domains. Its mathematical tractability, derived from its basis in the van der Pol oscillator, allows for analytical exploration of nonlinear dynamics that are prohibitively complex in more biophysically detailed models like the Hodgkin-Huxley equations [22]. This has enabled researchers to probe fundamental principles of excitability and rhythmicity not only in neuroscience but also in cardiology, endocrinology, and computational engineering.

Modeling Diverse Excitable Tissues

A primary application of the FHN model extends beyond its neuronal origins to the study of other electrically active biological tissues. Although usually discussed in the context of neuronal cells, action potentials also occur in many excitable cells such as cardiac muscle and some endocrine cells [21]. The model's two-variable reduction—one for a fast voltage-like variable and one for a slow recovery variable—effectively captures the essential regenerative and refractory properties common to these systems. For instance, in cardiac electrophysiology, the FHN framework has been adapted to study spiral wave dynamics and re-entrant arrhythmias, where the simplified phase plane analysis provides clear insights into wave initiation, propagation, and termination that are more obscured in high-dimensional ionic models. The analogy of an axon as "a piece of wire coated with gunpowder," where the gunpowder represents voltage-gated sodium channels enabling regenerative depolarization, is a conceptual bridge that the FHN model mathematically formalizes for diverse excitable membranes [21].

Investigating Specialized Sensory Neurons

Building on its role in general excitability, the FHN model has been instrumental in analyzing the response properties of specific neuronal subclasses, particularly sensory receptors. Here we examine a class of neurons that have been recently explored, the somatosensory neuronal subclass of cold thermosensors [8]. Researchers have employed modified FHN equations to analyze how temperature modulates firing thresholds and patterns in these neurons. A key finding from such modeling work is the quantitative description of how the model's parameters (e.g., the threshold parameter $a$ and timescale of the recovery variable $\epsilon$) must be temperature-dependent to replicate experimental data. This allows the model to simulate the dynamic range and sensitivity of cold receptors, providing a mathematical link between molecular transduction mechanisms, such as TRPM8 channel kinetics, and the resulting spike trains that encode thermal information for the nervous system [8]. This integration of genomics and mathematical modeling creates a powerful toolkit, as "with the progress being made in genomics and molecular genetics recently there has never been a better time to explore the nervous system and its disorders" through such interdisciplinary approaches [8].

Foundation for Advanced Neural Models and Neuromorphic Computing

The FHN model's simplicity makes it a foundational building block for more complex and biologically plausible computational neuroscience models. It sits conceptually between the abstract integrate-and-fire neuron and the detailed conductance-based models. While integrate-and-fire models are prized for analytical tractability in studying spike timing and network dynamics—since "information processing in the nervous system is carried out by spike timings in neurons" [10]—they often lack a dynamical representation of the spike generation mechanism itself [11]. Conversely, the FHN model explicitly includes this nonlinear spike generation, making it a preferred starting point for developing models that incorporate adaptation, bursting, or response to noisy inputs. For example, fractional calculus extensions of leaky integrate-and-fire models that capture multiple timescale dynamics cite the need to move beyond simple models because "the voltage trace of real neurons can follow multiple timescale dynamics... that arise from the interaction of multiple active membrane conductances" [23]. The FHN model, with its active, excitable dynamics, is a step in that direction. Furthermore, the FHN model's dynamics directly inspire the design of neuromorphic computing systems. Neuromorphic Spiking Neural Networks (SNNs) are promising computational paradigms that take deep inspiration from the working of mammalian brains, firing neurons, and synaptic plasticity [9]. The model's equations can be directly implemented in analog Very-Large-Scale Integration (aVLSI) circuits or with emerging nanodevices like memristors to create energy-efficient, brain-inspired hardware. In these implementations, the transistor or device characteristics are engineered to mimic the nonlinear nullclines of the FHN system. Such neurons are also used in implementing Recurrent Spiking Neural Networks (RSNNs) that can learn through hardware-friendly algorithms like e-prop (eligibility propagation) [9]. The local, online nature of learning rules like e-prop is compatible with the low-power, event-driven operation of FHN-based silicon neurons, enabling the development of autonomous systems for real-time signal processing and pattern recognition.

A Template for Nonlinear Dynamics in Other Fields

Beyond biology and computing, the FHN equations serve as a canonical model for excitable media in physics and chemistry. The system exhibits phenomena such as:

• Pulse propagation in active media
• Formation of spatial patterns like Turing patterns when coupled with diffusion terms
• Synchronization and wave phenomena in coupled oscillator networks

Its mathematical form, $\epsilon \frac{dv}{dt} = v - \frac{v^3}{3} - w + I_{\text{ext}}$ and $\frac{dw}{dt} = v + a - bw$, is studied in applied mathematics to understand bifurcations, canard cycles, and relaxation oscillations. This cross-disciplinary relevance underscores the model's power as a minimal representation of a broad class of systems where a fast, self-enhancing process (activation) is coupled with a slow, suppressing process (recovery). In summary, the FitzHugh-Nagumo model's applications span from explaining the threshold behavior of cold-sensitive neurons [8] to serving as the computational unit in next-generation neuromorphic chips [9]. Its enduring value lies in its ability to distill the universal essence of excitability into a pair of coupled differential equations, providing a versatile platform for exploration across the life sciences, physical sciences, and engineering.